Download Day 1 Notes: Introduction to Sequences and Series

Pre-Calculus
Semester 1
Created By: Jennifer Suby and Kay Knutson
Oswego East High School Fall 2015
Sequences &
Series
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Sequences and Series Day 1 Notes: Arithmetic Sequences
Sequences
Sequence: a function whose domain is a set of consecutive integers.
Domain: the position of the term (1st, 2nd, 3rd, and so on).
Range: the terms of the sequence.
Terms: the numbers in the sequence.
Finite Sequence: a sequence that has a last term.
Infinite Sequence: a sequence that continues without stopping.
Rule for the nth term: means to write an equation of the arithmetic sequence.
Look at the following sequences:
an = 2, 4, 6, 8
an = 2, 4, 6, 8, …..
If n represents the position of the term, then a2 refers to the 2nd term.
Writing Terms of Sequences
To write the first six terms of a sequence, plug the numbers 1 – 6 in for n.
Example 1: Write the first six terms of the sequence.
a) an = 5 – n
b) f(n) = 5n
Example 2: Write the next term in the sequence.
4 5 6
, , , _____,...
7 8 9
a) 1, 4, 9, 16, 25, ______, …
b) 5, 25, 125, ______, …
c)
d) 0, 3, 8, 15, 24, ______, …
e) 1, 2, 5, 10, 17, ______, …
f) 8, 16, 32, 64, ______
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Arithmetic Sequences
Common difference: in an arithmetic sequence, take the second term minus the first term. The common difference is denoted using
the letter d.
a1: represents the first term of a sequence.
The nth term of an arithmetic sequence with first term a1 , and common difference d is given by:
a n = a1 + (n − 1)d
or
=
an term value + ( n − term number ) d
To write the rule, replace a1 and d with what they equal and simplify or replace the term value, term number, and d with what they
equal and simplify.
Writing a Rule for the nth Term
Example 3: Write a rule for the nth term of the arithmetic sequence. Then find a12.
32, 47, 62, 77, …
Example 4: Write a rule for the nth term of the arithmetic sequence. Then find a25.
5 11 7 1
1
, , , , − ,
2 6 6 2
6
Finding the nth Term Given a Term and the Common Difference or Two Terms
If you are given two terms, in order to find the common difference, d, use the following formula: d =
right term value − left term value
big n − small n
Example 5: Write a rule for the nth term of the arithmetic sequence.
a) a8 = 50 and d = 0.25
b) d = –6 and a12 = –4
4
c) a 7 = 34 and a18 = 122
d) a1 = –2 and a9 = −
1
6
Example 6: Find the indicated term of each arithmetic sequence.
a) a1 = 76, a3 = 70, a101 = ?
b) a3 = 8, a5 = 14, a50 = ?
Example 7: How many terms are in the arithmetic sequence 18, 24, …, 336?
Example 8: Find the number of multiples of 7 between 30 and 300.
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Pre-Calculus
Sequences and Series Day 1: Arithmetic Sequences Worksheet
Name: _____________________________________
For numbers 1 – 3, write the first six terms of the sequence.
1. a n =
n2
n +1
3. a n = 3n 2 − 2n + 4
2. a n = 5n − 6
For numbers 4 – 6, decide whether the sequence is arithmetic. Explain why or not.
4. 3, 6, 12, 24, 48, …
5. 5, 8.4, 11.8, 15.2, …
6. –5, –3, –1, 1, 3, …
For numbers 7 – 9, write a rule for the nth term of the arithmetic sequence. Then find a30.
7. 5, 6, 7, 8, 9, …
8. 1, –4, –9, –14, –19
9. –4, –2, 0, 2, 4
For numbers 10 – 12, write a rule for the nth term of the arithmetic sequence.
10. d =
2
, a1 = 12
3
11. a8 = 36, a12 = 52
12. d = 11, a11 = 29
For numbers 13 and 14, find the indicated term of each arithmetic sequence.
13. a1 = 9, a2 = 17, a20 = ?
14. a7 = –3, a27 = 37, a1 = ?
15. How many terms are in the arithmetic sequence 178, 170, …, 2?
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Pre-Calculus
Sequences and Series Day 1: Arithmetic Sequences Homework
Name: _____________________________________
For numbers 1 – 3, write the first six terms of the sequence.
1. a n =
n+4
n
3. a n = 4n 2 − 3n − 4
2. an = (n + 1)3
For numbers 4 – 6, decide whether the sequence is arithmetic. Explain why or not.
4. 4, 7, 10, 13, 16, …
5. 2, 5, 8, 11, 14, …
6.
9 5 11 13
, , , 3, ,…
4 2 4
4
For numbers 7 – 9, write a rule for the nth term of the arithmetic sequence. Then find a30.
7. 1, 5, 9, 13, 17, …
8. –3, 0, 3, 6, 9, …
9. 2, 9, 16, 23, 30, …
For numbers 10 – 12, write a rule for the nth term of the arithmetic sequence.
10. d = 0.2, a1 = 16
11. a15 = –19, a24 = –16
12. d = 2, a6 = 15
For numbers 13 and 14, find the indicated term of each arithmetic sequence.
13. a1 = 15, a2 = 21, a20 = ?
14. a8 = 25, a20 = 61, a1 = ?
15. How many terms are in the arithmetic sequence 11, 16, …, 496?
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Sequences and Series Day 2 Notes: Arithmetic Series
Series
Series: when the terms of a sequence are added together.
Summation Notation: (sigma notation). Greek letter sigma: Σ
Think about Microsoft Excel. There is a button that looks like sigma on the toolbar. If you use it, it will ADD cells together.
Finite Series: 2 + 4 + 6 + 8
4
Summation notation: 2 + 4 + 6 + 8 =
∑ 2i .
i =1
Index of summation: i
Lower limit: 1 (the first position)
Upper limit: the last position (in the example it is 4)
4
∑ 2i
“The sum from i equals 1 to 4 of 2i”
i =1
Infinite Series: 2 + 4 + 6 + 8 + …
Summation notation is very similar. If the terms go on FOREVER, what math symbol does this refer to? [ ∞ ] infinity!
∞
So your upper limit changes to ∞.
∑ 2i
i =1
Writing Series with Summation Notation
*Find the pattern the SAME way as sequences.
*For finite series, determine the upper limit.
**NOTE! The index of summation does not have to be i – any letter can be used! The index does not have to start at 1. If it
starts at 3, it means that you are starting with the 3rd term.
Example 1: Write the series with summation notation.
a) 3 + 6 + 9 + 12
b) 7 + 2 – 3 – 8 – …
11
Example 2: Find the sum of the following series
10
5
a)
∑ 2i
2
− 1.
b)
∑i
2
+1
i =5
i =1
The expression formed by adding the terms of an arithmetic sequence is called an arithmetic series. The sum of the first n terms of an
arithmetic series is denoted by S n .
The sum of the first n terms of a finite arithmetic series is:
Sn
 ( a1 + an ) 
n=
Sn
or


2


(n (a
1
+ an ) )
2
In words, S n is the mean of the first and nth terms, multiplied by the number of terms.
Example 3: Consider the arithmetic series 20 + 18 + 16 + 14 + …
a) Find the sum of the first 25 terms.
b) Find n such that Sn = –760.
Example 4: Consider the arithmetic series 100 + 110 + 120 + 130 + …
a) Find the sum of the first 18 terms.
b) Find n such that Sn = 600.
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Example 5: The first row of a concert hall has 25 seats, and each row after the first has one more seat than the row before it. There
are 32 rows of seats.
a) Write a rule for the number of seats in the nth row.
b) Thirty-five students from a class want to sit in the same row. How close to the front can they sit?
c) How many seats are in rows 12 through 22?
Example 6: Find the sum of the series.
50
a)
∑ ( 4i + 3)
i =1
3
b)
∑ ( −3)
n
n=0
13
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Pre-Calculus
Sequences and Series Day 2: Arithmetic Series Worksheet
Name: _____________________________________
For numbers 1 – 3, write the series with summation notation.
1. 2 + 4 + 6 + 8 + 10
2. 1 – 5 – 11 – 17
3. 5 + 10 + 15 + 20 + …
For numbers 4 – 6, find the sum of the series.
14
4.
∑ ( 6 j − 10 )
j =1
4
60
5. ∑ (− 2 )n
6.
n =0
∑ (12 + 7i )
i =1
For numbers 7 and 8, for part (a), find the sum of the first n terms of the arithmetic series. For part (b), find n for the given sum Sn.
7. 40 + 37 + 34 + 31 + …
a) n = 20
b) Sn = –208
8. 1.50 + 1.45 + 1.40 + 1.35+ …
a) n = 15
b) Sn = 23.25
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Pre-Calculus
Sequences and Series Day 2: Arithmetic Series Homework
Name: _____________________________________
For numbers 1 – 3, write the series with summation notation.
1. 1 + 3 + 5 + 7
2. 100 + 110 + 120 + 130 + …
3. 5 – 6 – 17 – 28 – 39 – 50 – 61
For numbers 4 – 6, find the sum of the series.
50
4.
∑ (3.1i + 2.5)
i =1
5
[
5. ∑ 3 + (− 1)n
n =0
]
6.
25

1
i =1


∑  5i − 4 
For numbers 7 and 8, for part (a), find the sum of the first n terms of the arithmetic series. For part (b), find n for the given sum Sn.
7. 0.5 + 0.9 + 1.3 + 1.7 + …
a) n = 10
b) Sn = 189
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8. –6 + (–2) + 2 + 6 + …
a) n = 28
b) Sn = 442
9. To begin the half-time performance, a high school band marches onto the football field in pyramid formation. The drum major
leads the band alone in the first row. There are two band members in the second row, three in the third row, four in the fourth row,
and so on. The pyramid formation has 10 rows. How many members does the band have?
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Pre-Calculus
Sequences and Series Review Days 1 & 2
Name: _____________________________________
For numbers 1 – 3, write the first four terms of each sequence whose general rule is given.
1. an = 3n + 2
2. an =
2n
n+4
3. an =
(− 1)n+1
2n − 1
For numbers 4 – 6, find each indicated sum.
20
4. ∑ 5i − 7
i =1
50
4
6. ∑ k + 4
5. ∑ 2i 2
i =1
k =1
For numbers 7 – 9, express each sum using summation notation. Use 1 as the lower limit of summation and i for the index of
summation.
7. 9 + 17 + 25 …
8. –5 – 8 – 11 – 14
9. 1 + 2 + 3 + … + 30
For numbers 10 and 11, write the rule of the arithmetic sequence.
10. a6 = 300, d = 20
11. a2 = 2, a6 = 0
For numbers 12 – 14, find the indicated term of the arithmetic sequence with first term a1, and common difference d.
12. Find a6 when a1 = 13, d = 14.
13. Find a50 when a1 = 7, d = 5.
14. Find a60 when a1 = 35, d = –3.
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For numbers 15 & 16, find the indicated term of each arithmetic sequence.
15. a1 = 2, a2 = 22, a200 = ?
16. a9 = –38, a74 = –363, a1 = ?
17. Find the sum of the first 20 terms of the arithmetic sequence: 4, 10, 16, 22, …
18. Find the sum of the first 25 terms of the arithmetic sequence: 7, 19, 31, 43, …
19. Find n for the given sum Sn = 2272 of the arithmetic series: 9 + 13 + 17 + 21 + 25 + …
20. Find n for the given sum Sn = –138 of the arithmetic series: 5 + 2 + (–1) + (–4) + (–7) + …
21. How many terms are in the arithmetic sequence 2, –3, …, –813?
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Sequences and Series Day 3 Notes: Geometric Sequences and Series
Geometric Sequences
Common ratio: in a geometric sequence, take the second term divided by the first term. The common ratio is denoted using the
letter r.
a1: represents the first term of a sequence.
The nth term of an geometric sequence with first term a1, and common ratio r is given by:
an = (a1)(r)(n – 1)
or
an = ( term value )( r )
( n − term number )
To write the rule, replace a1 and r with what they equal and simplify or replace the term value, term number, and r with what they
equal and simplify.
Writing a Rule for the nth Term
Example 1: Write a rule for the nth term of the geometric sequence. Then find a8.
5, 2, 0.8, 0.32, …
Finding the nth Term Given a Term and the Common Ratio or Two Terms
If you are given two terms, in order to find the common ratio, r, use the following formula: r = big n −small n
right term value
left term value
Example 2: Write a rule for the nth term of the geometric sequence.
a) a4 = 3 and the common ratio r = 3.
b) a2 = –18 and a5 =
2
3
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c) a2 = –4 and a6 = –1024
Geometric Series
The expression formed by adding the terms of a geometric sequence is called a geometric series. The sum of the first n terms of a
geometric series is denoted by Sn.
The sum of the first n terms of a finite geometric series with common ratio r ≠ 1 is:
( a (1 − ( r ) ))
=
n
1
Sn
(1 − ( r ) )
Example 3: Consider the geometric series
2 – 8 + 32 – 128 + …
a) Find the sum of the first 10 terms.
b) Find n such that Sn = 6,554
22
Example 4: Consider the geometric series:
12 + 24 + 48 + 96 + ...
b) Find n such that Sn = 12,276
a) Find the sum of the first 8 terms.
Example 5: Find the indicated term of each geometric sequence.
a) a1 = 2– 4, a2 = 2– 3, a12 = ?
b) a2 = 64, a5 = –8, a9 = ?
 4
Example 6: Find the sum of the geometric series: ∑ 5  − 
5
i =1 
12
i −1
23
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Pre-Calculus
Sequences and Series Day 3: Geometric Sequences and Series Worksheet
1. Find the common ratio of the geometric sequence: − 2 ,
Name: _____________________________________
1 1 1
, − , , ...
2 8 32
2. Write a rule for the nth term of the geometric sequence. Then find a8.
2, –0.8, 0.32, –0.128, …
3. Write a rule for the nth term of the geometric sequence: a3 = –64, a7 = −
1
4
4. For part (a), find the sum of the first n terms of the geometric series. For part (b), find n for the given sum Sn.
2 – 6 + 18 – 54 + …
b) Sn = 1094
a) n = 19
12
5. Find the sum of the geometric series:
∑ − 5(3)
i −1
i =1
3 3 3
6. Decide whether the sequence is arithmetic, geometric, or neither: − 3, ,− , ,...
4 16 64
7. Find the sum of the first 11 terms of the geometric series: 3 – 6 + 12 – 24 …
For numbers 8 and 9, find the indicated term of each geometric sequence.
8. a1 = 2, a2 = 2
3
2
, a13 = ?
9. a2 = 81, a5 = 24, a7 = ?
25
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Pre-Calculus
Sequences and Series Day 3: Geometric Sequences and Series Homework
Name: _____________________________________
For numbers 1 and 2, find the common ratio of the geometric sequence.
1. 1, –3, 9, –27, …
2. 7 ,
21 63 189
, ...
, ,
4 16 64
For numbers 3 and 4, write a rule for the nth term of the geometric sequence. Then find a8.
3 3 3
3. 3, , , ,...
2 4 16
4. 7, 28, 112, 448, …
For numbers 5 and 6, write a rule for the nth term of the geometric sequence.
5. a1 = 5, r = 1.1
6. a8 =
1
, a15 = 243
9
7. For part (a), find the sum of the first n terms of the geometric series. For part (b), find n for the given sum Sn.
–20 + 40 – 80 + 160 – …
a) n = 10
b) Sn = –873,820
27
For numbers 8 and 9, find the sum of the geometric series.
10
8.
∑
2(8)
i −1
i =1
3
3 
∑
i =1  2 
5
9.
i −1
For numbers 10 and 11, decide whether the sequence is arithmetic, geometric, or neither.
10. 1, –2, –5, –8, …
11.
1 2 4 8
, , , ,...
3 9 27 81
12. Find the sum of the first 12 terms of the geometric series: 2 + 6 + 18 + 54 + …
13. Find the sum of the first 12 terms of the geometric series: −
3
+ 3 − 6 + 12 − ...
2
For numbers 14 and 15, find the indicated term of each geometric sequence.
14. a1 = 2, a9 =13,122, a12 = ?
15. a2 = 1, a7 =
1
, a9 = ?
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Pre-Calculus
Sequences and Series Review Days 1 – 3
Name: _____________________________________
an = ( term value )( r )
=
an term value + ( n − term number ) d
n − term number
Sn
(n (a
=
1
+ an ) )
2
( a (1 − ( r ) ))
=
n
1
Sn
(1 − ( r ) )
For numbers 1 – 3, write the next term in the sequence.
1. 5, 8, 11, 14, 17, ______, …
2. 3, 9, 27, 81, ______, …
3.
2 2 6
, , , 1, _____, ...
5 3 7
For numbers 4 – 7, find the sum of the series.
6
4.
∑ 5k
k =0
4
5.
 1
∑  − 3 
i
∑ (2i
5
6.
2
)
−3
i =1
i=2
7
7.
∑ 12
m =3
For numbers 8 – 13, write a rule for the nth term of the arithmetic sequence.
8. 4, 7, 10, 13, 16, …
9. a1 = –3 ; d = 4
10. a1 = 2 ; d = –2
11. a5 = 11; a11 = 29
12. a4 = 18; a10 = 48
13. a7 = –22; a11 = –34
14. How many terms are in the arithmetic sequence −
21 9
141
?
, − ,...,
4
2
2
For numbers 15 and 16, find the indicated term of each arithmetic sequence.
15. a1 = 12.5, a2 = 13, a147 = ?
16. a15 = 186, a27 = 342, a12 = ?
29
17. Find the sum of the first 20 terms of the arithmetic series: 2 + 3 + 4 + 5 + 6 + ….
18. Use the series from #17 to find n when Sn = 5150.
19. Find the sum of the first 15 terms of the arithmetic series: 25 + 35 + 45 + 55 + ….
20. Use the series from #19 to find n when Sn = 3105.
For numbers 21 – 24, write a rule for the nth term of the geometric sequence.
21. –5, 10, –20, 40, …
22. a1 = 1; r =
3
5
23. a3 = 18; a6 = –486
24. a3 = 180; a6 =38,880
25. Find the sum of the first 8 terms of the geometric series: 1 – 4 + 16 – 64 + …
26. Use the series in #25 to find n when Sn = –819
27. Find the number of multiples of 6 between 28 and 280.
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Sequences and Series Day 4 Notes: Infinite Geometric Series
The Sum of an Infinite Geometric Series
The first term is a1 and the common ratio is r, to find the sum of an infinite geometric series use the following formula: S =
a1
1− r
provided | r | < 1.
A series converges if it does have a sum.
If | r | ≥ 1, the series has no sum. A series diverges if it does not have a sum.
Example 1: Find the sum of the infinite geometric series, if possible.
∞
a)
∑ 2(0.1)
i −1
b) 12 + 4 +
i =1
∞
c)
∑
∞
4(0.6 )n
d)
n =0
e) − 30 + 15 −
4 4
+ ...
3 9
5
4 
∑
n =1  4 
n −1
15 15
+ ...
2 4
Finding the Common Ratio
Given the sum S and first term a1, plug in what you know and solve for r!
Example 2: Find the common ratio of the infinite geometric series with the given sum and first term.
a) S =
27
; a1 = 5
5
b) S = –2; a1 = –
4
3
31
c) S =
2
8
; a1 =
9
3
Finding the First Term
Given the sum S and common ratio r, plug in what you know and solve for a1!
Example 3: Find the first term of the infinite geometric series with the given sum and common ratio.
a) S = 54 ; r = 0.2
c) S =
b) S = 2; r =
1
3
2
8
;r=
9
3
Writing a Repeating Decimal as a Fraction
1) Find the repeating portion. (it could be 1 digit, 2 digits, 3 digits, etc…)
2) Create a fraction with a denominator of 9’s. (Use as many 9’s as there are repeating digits)
3) Simplify.
4) If there are terms on the left side of the decimal you must use MIXED fractions.
***If there are terms that DON’T repeat, you need to move those numbers to the left of the decimal place, then use mixed fractions.
Remember to move the decimal of your denominator back the amount you moved the numbers over. (See example d)
Example 4: Write the repeating decimal as a fraction.
a) 0.6666…..
b) 0.327327…
c) 27.2727…
d) 0.416666…
32
Pre-Calculus
Sequences and Series Day 4: Infinite Geometric Series Worksheet
Name: _____________________________________
For numbers 1 – 4, find the sum of the infinite geometric series if it has one.
5
∑ 2 4 
n =0  
∞
1.
n
∞
2.
∑ 0.6(0.1)
∞
n
3.
∑ 4(0.6)
n
n =0
n =0
17
∑ 48
 
n =1
∞
4.
n −1
For numbers 5 – 7, find the common ratio of the infinite geometric series with the given sum and first term.
5. S =
1
8
, a1 =
4
9
6. S =
10
, a1 = 1
9
7. S = 200, a1 = 100
For numbers 8 – 10, write the repeating decimal as a fraction.
8. 0.4040 …
11. Find the sum of the infinite geometric series: 1 −
9. 0.543543 …
10. 150.150150 …
1 1 1
+ − + ...
2 4 8
33
34
Pre-Calculus
Sequences and Series Day 4: Infinite Geometric Series Homework
Name: _____________________________________
For numbers 1 – 5, find the sum of the infinite geometric series if it has one.
1.
 1
∑ 4 − 3 

n =1 
4.
7
∑ 3 2 
n =1  
∞
∞
n −1
n
2.
1
∑ − 7 3 
 
n =0
5.
1 1
∑ − 6 − 2 


n =1
∞
n −1
∞
1
∑ 0.3 10 
 
n =1
∞
3.
n −1
n −1
For numbers 6 – 8, find the common ratio of the infinite geometric series with the given sum and first term.
6. S = 4, a1 = 7
7. S =
16
, a1 = 8
3
8. S = −
44
, a1 = –4
15
For numbers 9 – 11, write the repeating decimal as a fraction.
9. 0.888 …
10. 27.2727 …
11. 0.653653 …
For numbers 12 and 13, find the sum of each infinite geometric series.
1 1 1
+ ...
12. 1 + + +
3 9 27
1 1
13. 3 − 1 + − + ...
3 9
35
36
Pre-Calculus: Sequences & Series
Name: _____________________________________
Find the next term of each sequence, then state whether the above sequences are arithmetic, geometric or neither.
1. 20, 18, 16, 14, _____
12. 1, 3, 7, 15, 31, 63, _____
2. 1, 2, 4, 8, 16, 32, _____
13. 1, 3, 9, 27, 81, 243, _____
3. 1, 3, 6, 10, 15, 21, _____
14. 1, 10, 100, 1000, _____
4. 1, 4, 9, 16, 25, 36, _____
15.
5. 2, 6, 15, 31, 56, 92, _____
16. 3, –12, 48, –192, 768, _____
6. T, Q, N, K, H, E, _____
17. 1, 5, 17, 53, 161, 485, _____
7.
1 1 1 2
, , , , _____
6 3 2 3
8. 2, –3, –8, –13, _____
9. 1,
5 3 7
, , , _____
4 2 4
1 1 1 1
, , ,
, _____
2 4 8 16
18. 0, 3, 8, 15, 24, 35, _____
19. 1,
3 9 27
, ,
, _____
2 4 8
20. 10, 21, 32, 43, 54, _____
10. A, C, F, J, O, _____
21. 0, 10, 21, 33, 46, 60, _____
11. 1, 1, 2, 3, 5, 8, 13, _____
22. 1, 2, 5, 14, 41, 122, _____
CHALLENGE:
a) 18, 46, 94, 63, 52, 61, _____
c) 6, 8, 5, 10, 3, 14, 1, _____
b) O, T, T, F, F, S, S, E, N, _____
d) A E F H I K L M N T V W
B C D G J O P Q R S U
Where do the X, Y, and Z go?
37
38
Pre-Calculus
Sequences and Series Multiple Choice Packet
Name: _____________________________________
Day 1:
1. What is the third term of the sequence defined by a n = 4n + 6 ?
a) 6
b) 10
c) 14
d) 18
e) 22
d) 20
e) 26
d) 30
e) 32
c) 23
d) 25
e) 27
c) –5, –2, 1, 4, 7, …
d) –3, 0, 4, 9, 15, …
e) 3, 5, 8, 12, 17, …
c) 3, 9, 21, 39, 63, …
d) –3, 0, 6, 15, 27, …
e) 3, 8, 13, 18, 23, …
2. What is the fifth term of the sequence defined by a n = 3n − 1 ?
a) 2
b) 8
c) 14
3. What is the next term in the sequence 1, 6, 11, 16, 21, …?
a) 23
b) 26
c) 28
4. What is the next term in the sequence 4, 7, 10, 13, 16, …?
a) 19
b) 21
5. Which of the following is an arithmetic sequence?
a) 2, 5, 9, 14, 20, …
b) 1, 3, 6, 10, 15, …
6. Which of the following is an arithmetic sequence?
a) 2, 4, 8, 14, 22, …
b) 1, 5, 6, 10, 11, …
7. What is a rule for the nth term of the arithmetic sequence with a10 = 22 and common difference d = 3?
a) a n = 3n − 2
b) a n = 3n − 8
c) a n = 3n + 4
d) a n = 3n − 6
e) a n = 3n + 5
8. What is a rule for the nth term of the arithmetic sequence with a21 = 147 and common difference d = 11?
a) a n = 11n − 21
b) a n = 11n − 42
c) a n = 11n + 21
d) a n = 11n + 32
e) a n = 11n − 84
9. What is a rule for the nth term of the arithmetic sequence with a8 = 21 and a14 = 45?
a) a n = 4n − 11
b) a n = 4n − 8
c) a n = 4n + 1
d) a n = 4n + 7
e) a n = 4n + 11
c) 13 + 18 + 23
d) 8 + 11 + 14
e) 8 + 13 + 18
c) 3 + 7 + 11
d) 10 + 14 + 18
e) 10 + 21 + 32
c) 1860
d) 2010
e) 2150
c) 7, 12, 17, 22, 27, …
d) –4, 0, 4, 8, 12, …
e) –6, –4, –2, 0, 2, …
Day 2:
3
10. Which series is represented by
∑ (5i + 3) ?
i =1
a) 8 + 13 + 18 + …
b) 3 + 8 + 13
∑ (2k
4
11. Which series is represented by
2
)
+k ?
k =2
a) 3 + 10 + 21
b) 10 + 21 + 36
12. What is the sum of the first 20 terms of the series: 3 + 14 + 25 + 36 + . . . ?
a) 226
b) 1252
Day 3:
13. Which of the following is a geometric sequence?
a) 1, 2, 3, 4, 5, …
b) 1, 3, 9, 27, 81, …
39
14. Which of the following is a geometric sequence?
a) 1, 2, 4, 8, 16, …
b) –2, –4, 8, 16, –32, …
c) 4, 8, 24, 96, 480, …
d) –5, 0, 10, 25, 45
e) –3, 1, 5, 9, …
15. What is a rule for the nth term of the geometric sequence –3, –6, –12, –24, –48, …?
a) a n = 2(− 3)
n −1
b) a n = −3(2 )
c) a n = 3(− 2 )
n −1
d) a n = −3(− 2 )
n −1
n −1
e) a n = −2(3)
n −1
16. What is a rule for the nth term of the geometric sequence with a4 = –18 and common ratio r = 2?
a) a n = 2.25(2 )
n −1
b) a n = 2(2.25)
c) a n = −2(2.25)
n −1
n −1
d) a n = −2.25(2)
n −1
e) a n = −2.25(− 2 )
n −1
17. What is a rule for the nth term of the geometric sequence with a2 = 4 and a6 = 2500?
a) a n = 0.8(5)
n −1
b) a n = 4(5)
c) a n = −0.8(5)
n −1
n −1
d) a n = −4(5)
n −1
e) a n = 5(0.8)
n −1
18. What is the sum of the first 12 terms of the geometric series 1 + 2 + 4 + 8 + 16 + …?
a) 240
b) 880
c) 1060
d) 1850
e) 4095
Day 4:
19. What is the sum of the series: 1 −
a)
4
9
b)
1 1 1
+ −
+ ... ?
3 9 27
2
3
c)
∞
20. What is the sum of the series:
∑ 3(0.2)
i −1
3
4
d)
5
6
e)
7
8
?
i =1
a) 2.25
b) 2.75
c) 3.25
d) 3.75
e) 4.25
21. What is the common ratio of an infinite geometric series whose sum is 30 and the first term is a1 = 6?
a)
2
3
b)
1
5
c)
3
4
d)
4
5
e)
4
7
22. What is the common ratio of an infinite geometric series whose sum is 125 and the first term is a1 = 625?
a) –4
b) –2
c) −
1
2
d) 2
e) 4
23. What fraction is equivalent to the repeating decimal 4.54545…?
a)
29
7
b)
81
19
c)
75
17
d)
50
11
e)
125
26
40
Pre-Calculus
Sequences and Series Unit Review Sheet
Name: _____________________________________
For numbers 1 – 3, decide whether the sequence is arithmetic, geometric, or neither. If it is arithmetic or geometric find the rule for
the nth term.
1 1 1 1
2. –1, –4, –7, –10, –13, …
3. 1, 2, 4, 8, 16, …
1. 1, , , , ,...
2 4 8 16
For numbers 4 and 5, write the terms of the series. Then evaluate the sum.
5
4.
7
∑ (n − 5)
5.
n =1
∑ (i − 1)
2
i =3
For numbers 6 and 7, evaluate the sum of the arithmetic series.
25
6.
40
∑ (4i )
7.
i =1
∑ (3i − 10)
i =1
For numbers 8 – 10, evaluate the sum of the finite or infinite geometric series. (The infinite geometric series problems may not have
a sum.)
12
8.
∑ (3)(2)n−1
n =1
∞
9.
3
∑ (− 1) 2 
n =1
n −1
10.
 
∞
 1  3 
n =1
  
∑  2  4 
n −1
For numbers 11 and 12, write the first five terms of the following sequences.
11. a1 = 6, r =
1
2
12. a1 = 7, d = –12
13. Write a rule for the nth term of the arithmetic sequence: a5 = 18, a17 = 66
41
14. Find a84 for the arithmetic sequence: 12, 6, 0, ….
15. Find S84 for the arithmetic series 12 + 6 + 0 + ….
16. Find n if Sn = –10,602 for the arithmetic series 12 + 6 + 0 + ….
For numbers 17 and 18, determine if the infinite series has a sum. Justify your response.
51
∑ 89
 
n =1
∞
17.
n −1
 8
∑ 7 − 7 

n =1 
∞
18.
n −1
19. Write the repeating decimal as a fraction: 0.5833
20. Find the sum of the first 15 terms of the geometric series: 5 – 15 + 45 – 135 + …
21. Use the series in #20 to find n when Sn = –8200
22. Write a rule for the nth term of the geometric sequence: a2 = 6, a5 = 162
42
23. In a geometric sequence, a2 = 64 and a5 = –8. Find a9.
24. In a arithmetic sequence a3 = 8 and a5 = 14. Find a50.
25. Consider the sequence of all multiples of 3 between 1 and 200.
b) Find the sum of the sequence.
a) Find the rule for an.
Numbers 26 – 32 are multiple choice.
26. Which of the following series converges?
a)
1 3 9 27
+ + +
+ ...
2 2 2 2
b) –40 – 20 – 10 – 5 – …
d) 100 – 110 + 121 – 133.1 + …
c) 10 + 15 + 22.5 + 33.75 + …
e) 50 + 100 + 200 + 400 + …
27. An infinite geometric series has a sum of 200 and a common ratio of 0.4. What is the first term of the series?
a) 80
b) 120
c) 333
d) 500
e) 650
d) 27
e) 32
28. What is S6 for the arithmetic series 4 + 4.2 + 4.4 + 4.6 + 4.8 + …?
a) 5
b) 15
c) 20
29. What is the common ratio of the geometric sequence
a)
1
3
b)
1
2
3 3
, , 3, 6 , ...?
4 2
c) 2
d)
30. What is the approximate sum of the geometric series ∑ 3(0.9 )
8
5
2
e) 3
m −1
m =1
a) 6
b) 9
c)
19
2
d) 12
e) 17
43
31. Which is summation notation for the series 2 + 4 + 8 + 16?
4
a)
∑ 2k
k =1
4
4
b)
∑2
k =0
k
c)
∑k
4
2
k =1
d)
∑ 2k
k =0
4
e)
∑2
k
k =1
32. What is the common difference of the sequence 3, 4.5, 6, 7.5, …?
a) 1.5
b) 2
c) 3
d) 4.5
e) 5
44
Combinatorics
45
46
Combinatorics Day 1 Notes: Venn Diagrams
Venn diagram: a diagram used to represent a relationship between sets.
When drawing a Venn diagram, we begin with a rectangle, which represents a universal set U. Inside the rectangle we draw circles to
represent subsets of the universal set. For example, consider the Venn diagram below. Here we have a universal set of 1,000 typical
Americans. The set A represents those who own a dog. The set B represents those who own a cat. Of our 1,000 typical Americans, a
total of 450 are in set A and a total of 140 are in set B.
U
A
410
B
40
100
450
Notice that sets A and B overlap. This represents the fact that 40 of our 1,000 typical Americans have both a cat and a dog. We call
the set of elements that any two sets A and B have in common the intersection of A and B, which we denote by A ∩ B. It is the shaded
region in the Venn diagram below.
U
A
B
The set of people who have either a cat or a dog represents the union of sets A and B, which we denote by A ∪ B . It is the shaded
region in the Venn diagram below.
U
A
B
The set of all elements not in a set A is called the complement of A and is denoted A′. In the union diagram directly above, the
un-shaded region inside the rectangle represents A′ ∩ B′, the complement of A ∪ B . The complement consists of people who have
neither a dog or a cat.
Look again at the very first Venn diagram. If we let n( A ∪ B ) designate the number of elements in the union of sets A and B, we see
that n( A ∪ B ) = 550. Notice that
n( A ∪ B ) ≠ n(A) + n(B)
(that is, 550 ≠ 450 + 140), because the number of people in A ∩ B are counted twice when n(A) and n(B) are added. To compensate,
we must subtract n(A ∩ B) once from the sum:
n( A ∪ B ) = n(A) + n(B) – n(A ∩ B)
The intersection of events A and B is the empty set because they do not have anything in common. The symbol used to denote the
empty set is Ø. (The empty set is an event with nothing in it).
U
A
B
Example 1: Of the 540 seniors at Oswego East High School, 335 are taking mathematics, 287 are taking science, and 220 are taking
both math and science. Use a Venn diagram to determine how many are taking neither math nor science.
47
Example 2: In a survey, 113 business executives were asked if they regularly read the Wall Street Journal, Business Week magazine,
and Time magazine. The results of the survey are as follows:
88 read the Journal.
76 read Business Week.
85 read Time.
59 read the Journal and Business Week
6 read only the Journal.
5 read only Business Week.
8 read only Time.
42 read all three.
How many read none of the three publications?
Example 3: Of the 46 students in Student Council, 22 said they like cotton candy, 19 said they like caramel corn, and 17 said they
don’t like either. How many like cotton candy but not caramel corn?
Example 4: A certain small college has 1,000 students. Let F = the set of college freshmen, and let M = the set of music majors.
These sets are shown in the Venn diagram below. Describe each of the following sets in words and tell how many members it has.
U
F
285
a) F ∩ M
b) F ∪ M
c) F ′
d) M ′
e) F ′ ∩ M
f) F ∩ M ′
g) F ′ ∩ M ′
h) F ′ ∪ M ′
i) F ′ ∪ Ø
M
15
50
650
Example 5: Let U be the set of all letters from A to Z. Let A = {A, E, I, O, U} and let B = {B, E, F, I, J, K, R, T, Z}. Find the
indicated set.
a) A ∪ B
b) A ∩ B
c) B´
d) A ' ∩ B '
48
Pre-Calculus
Combinatorics Day 1 Worksheet
Name: _____________________________________
1. Let U be the set of all integers from 1 to 10. Let A = {1, 2, 4, 8} and let B = {2, 4, 6, 8, 10}. Find the indicated set.
a) A ∪ B
b) A ∩ B
c) A´
d) ( A ∪ B ) ´
2. Consider the sets defined below. Find the indicated set.
U = the set of all 12 months.
X = the set of all 30 day months.
Y = the set of all 31 day months
Z = the set of all months ending with “r”
a) X ∪ Z
b) X ∩ Z
c) Z´
d) ( X ∪ Z ) ´
3. Stephen asked 100 coffee drinkers whether they like cream or sugar in their coffee. According to the Venn diagram below, how
many like
a) Cream?
b) Sugar?
c) Sugar but not cream?
d) Cream but not sugar?
e) Cream and sugar?
f) Cream or sugar?
4. Ben asked 60 students whether they listen to two popular radio stations, WROK and WRAP. He found that 23 listen to WROK, 18
listen to WRAP, and 8 listen to both. How many students in Ben's survey listen to
a) WROK but not WRAP?
b) WRAP but not WROK?
c) neither WROK nor WRAP?
5. Oshkosh did a study of the colors used in African national flags. He found that 38 flags have red, 20 have blue, 13 have both red
and blue, and 8 have neither red nor blue. How many flags
a) have red but not blue?
b) have blue but not red?
c) were included in the study?
49
6. Kaleb asked 100 adults whether they had studied French, Spanish or Japanese in school. According to the Venn diagram below,
how many had studied
a) Spanish?
b) Spanish but not French?
c) Japanese but not French?
d) French and Spanish?
e) French or Spanish?
f) French and Spanish but not Japanese?
7. Coach Krutch offered to buy hot dogs for players on his team. Of the 44 players, 28 wanted ketchup, 20 wanted mustard, 14 wanted
relish, 10 wanted ketchup and mustard, 11 wanted ketchup and relish, 8 wanted mustard and relish and 6 wanted all three condiments.
How many players wanted
a) Ketchup only?
b) Mustard but not relish?
c) Relish but not mustard?
d) Ketchup and mustard but not relish?
e) Relish and mustard but not ketchup?
f) None of the three condiments?
8. Where does each of the following numbers belong on the Venn diagram?
2, 3, 4, 5, 6, 7, 8, 9, 10, 11.
Prime
Numbers
Even
Numbers
Multiples
of 3
9. 100 people were asked if they liked Math, Science, or Social Studies. Everyone answered that they liked at least one.
56 like Math
43 like Science
35 like Social Studies
18 like Math and Science
10 like Science and Social Studies
12 like Math and Social Studies
6 like all three subjects
a) How many people like Math only?
b) How many people like Science only?
c) How many people like Social Studies only?
d) How many people like at least two of the three?
50
Pre-Calculus
Combinatorics Day 1: Venn Diagrams Homework
Name: ______________________________
For numbers 1 and 2, draw a Venn diagram and shade the region representing the set given in part (a). Then draw a separate Venn
diagram and shade the region representing the set given in part (b).
1. a) P ∩ Q
b) P ∪ Q
2. a) (P ∪ Q ) ′
b) P ′ ∩ Q ′
For numbers 3 – 6, draw a Venn diagram to illustrate each situation described. Then use the diagram to answer each question asked.
3. In an election-day survey of 100 voters leaving the polls, 52 said they voted for Proposition 1, and 38 said they voted for
Proposition 2. If 18 said they voted for both, how many voted for neither?
4. Although the weather was perfect for the beach party, 17 of the 30 people attending got a sunburn and 25 people were bitten by
mosquitoes. If 12 people were both bitten and sunburned, how many had neither affliction?
5. In a survey of 48 high school students, 20 liked classical music and 16 liked bluegrass music. Twenty students said they didn’t like
either. How many liked classical but not bluegrass?
6. Of the 52 teachers at Roosevelt High School, 27 said they like to go sailing, 25 said they like to go fishing, and 12 said they don’t
enjoy either recreation. How many enjoy fishing but not sailing?
7. In a parking lot containing 85 cars, there are 45 cars with automatic transmissions, 43 cars with rear-wheel drive, and 46 cars with
four-cylinder engines. Of the cars with automatic transmissions, 26 also have rear-wheel drive. Of the cars with rear-wheel drive, 29
also have four-cylinder engines. Of the cars with four-cylinder engines, 27 also have automatic transmissions. There are 21 cars with
all three features.
a) How many cars do not have automatic transmissions and rear-wheel
drive but do have four-cylinder engines?
b) How many cars do not have any of the three features?
51
52
Combinatorics Day 2 Notes: The Fundamental Counting Principle and Permutations
Fundamental Counting Principal
TWO EVENTS: If one event can occur in m ways and another event can occur in n ways, then the number of ways that both events can
occur is mn.
(If one event can occur in 3 ways and another event can occur in 6 ways, then both events can occur in 3  6 = 18 ways.)
THREE OR MORE EVENTS: The fundamental counting principle “extended”. If 3 events can occur in m, n, and p ways, then the
number of ways that all three events can occur is mnp.
(If three events can occur in 3, 4, and 8 ways, then all three events can occur in 348 = 96 ways.)
Example 1:
a) At Oswego East High School, there are 273 freshmen, 291 sophomores, 252 juniors, and 292 seniors. In how many different ways
can a committee of 1 person from each class be chosen?
b) Mr. and Mrs. Cal Q. Leight go out to dinner after a long day sitting on their elliptical deck and enjoying a swim in their pool. At
the restaurant they go to, they each have a choice of 8 different entrees, 2 different salads, 12 different soft drinks, and 6 different
desserts. In how many ways can Mr. C choose 1 salad, 1 entrée, 1 soft drink, and 1 dessert?
Fundamental Counting Principle with Repetition
Example 2: The standard configuration of an Illinois license plate used to be 3 letters followed by 3 digits.
a) How many different license plates were possible if letters and digits can be repeated?
b) How many different license plates are possible if letters and digits cannot be repeated?
Example 3: A multiple choice test has 10 questions with 4 multiple choices for each question. In how many different ways could you
complete the test?
Permutations
Permutation: an ordering of n objects, where the order of the objects is important (does matter).
***There are 6 permutations of the letters A, B, and C: ABC, ACB, BCA, BAC, CAB, and CBA.
Since there are 3 choices for the 1st letter, 2 choices for the 2nd, and 1 choice for the 3rd, there are 3∙2∙1 = 6 ways to arrange the letters.
(3∙2∙1 can be written 3!)
In general, the number of permutations of n objects is: n! (Use this when ALL objects are being chosen)
53
Example 4: 10 skiers are competing in the final round of the Olympic freestyle skiing aerial competition.
a) In how many different ways can the skiers finish the competition? (Assume there are no ties)
Example 5: You have homework assignments from 5 different classes to complete this weekend. In how many different ways can
you complete the assignments, if the order is important?
Example 6: There are 8 movies you would like to see that are currently showing in theaters, you movie-goer! In how many different
ways can you see all of the 8 movies, if the order is important?
Permutations of n objects taken r at a time
The number of permutations of r objects taken from a group of n distinct objects is denoted by: n Pr =
n!
(Use this when
( n − r )!
only SOME of the objects are being chosen)
Example 7: 10 skiers are competing in the final round of the Olympic freestyle skiing aerial competition. In how many different
ways can 3 of the skiers finish first, second, and third place to win the gold, silver, and bronze?
Example 8: You have homework assignments from 5 different classes to complete this weekend. In how many different ways can
you choose 2 of the assignments to complete first and last?
54
Example 9: There are 9 players on a baseball team. In how many ways can you choose the batting order for all 9 of the players?
Example 10: How many 9-letter “words” can be formed using the letters of the word FISHERMAN? (Note: We allow any
arrangement of letters, such as “HAMERSNIF,” to count as a “word.” We also assume each letter is used exactly once).
Example 11: a) How many 4-digit numbers contain no 5’s?
b) How many 4-digit numbers contain at least one 5?
Permutations with Repetition (Keywords: distinguishable, identical)
Permutations that are not distinguishable, meaning you can’t tell one permutation or arrangement from another one in the list, are
often found with repeating objects…like the letters in the word WOW.
WOW
OWW
WWO
WOW
OWW
WWO
Only 3 are distinguishable without color (WOW, OWW, WWO)
So in this case the # of permutations is 3! / 2! = 3
In general, the number of distinguishable permutations of n objects where one object is repeated q1 times, another is repeated q2 times,
and so on is:
n!
( q1!⋅ q2!⋅ ... ⋅ qk!)
Example 12: Find the number of distinguishable permutations of the letters.
a) SUMMER
b) WATERFALL
c) ILLINOIS
Example 13: A person walks from A to B always traveling east or south. How many paths are possible?
A●
●B
55
Circular Permutations
To determine the number of circular permutations with n objects use (n – 1)!.
Example 14: How many circular permutations are possible when seating four people around a table?
56
Pre-Calculus
Combinatorics Day 2 Worksheet
Name: _____________________________________
Each event can occur in the given number of ways. Find the number of ways all of the events can occur.
1. Event 1: 2 ways, Event 2: 4 ways, Event 3: 5 ways, Event 4: 3 ways
For numbers 2 and 3, for the given configuration, determine how many different 5-digit postal zip codes are possible if (a) digits can
be repeated, and (b) digits cannot be repeated.
2. Has all even digits.
3. Begins with a 3 or a 1.
For numbers 4 and 5, evaluate the factorial.
4. 0!
5. 12!
For numbers 6 and 7, find the number of permutations.
6. 8P8
7. 10P1
For numbers 8 – 11, find the number of distinguishable permutations of the letters in the word.
8. PAPER
9. ALASKA
10. PERMUTATION
11. BILLIONAIRES
Use the following information for numbers 12 and 13.
Eight students are to be seated in a classroom with 11 desks.
12. Calculate the number of seatings by choosing one of the desks for each student.
13. Calculate the number of seatings by choosing one student for each of the desks, after increasing the number of students to 11 by
imagining that there are 3 “invisible” students (who are, of course, indistinguishable). Do you get the same answer as in #12?
14. A person walks from P to Q always traveling east or south. How many paths are possible?
P
•
•
Q
15. How many circular permutations are possible when seating ten people around a table?
57
58
Pre-Calculus
Combinatorics Day 2 Homework
Name: _____________________________________
Each event can occur in the given number of ways. Find the number of ways all of the events can occur.
1. Event 1: 6 ways, Event 2: 5 ways, Event 3: 12 ways
For numbers 2 and 3, for the given configuration, determine how many different 5-digit postal zip codes are possible if (a) digits can
be repeated, and (b) digits cannot be repeated.
2. Begins with 4.
3. Is divisible by 2.
For numbers 4 and 5, evaluate the factorial.
4. 4!
5. 11!
For numbers 6 and 7, find the number of permutations.
6. 12P0
7. 6P3
For numbers 8 – 11, find the number of distinguishable permutations of the letters in the word.
8. CHEMISTRY
9. EEL
10. SUCCESS
11. MATHEMATICS
12. In a dog show, how many ways can four Pomeranians, five golden retrievers, two Great Pyrenees, and six English terriers line up
in front of the judges if the dogs of the same breed are considered identical? In how many different ways can three dogs win first,
second, and third place?
13. You have forgotten the combination of the lock on your school locker. There are 40 numbers on the lock, and the correct
combination is “R_____-L_____-R_____.” How many possible combinations are there?
14. How many circular permutations are possible when seating five people around a table?
15. A photographer lines up the 13 members of family in a single line in order to take a photograph. How many different ways can the
photographer arrange the family members for the picture?
59
16. A discussion panel consisting of 4 women and 3 men is to be seated behind a long table at an open town meeting. In how many
ways can the panel be seated if men and women must be placed in alternate seats?
17. The window of a music store has 8 stands in fixed positions where instruments can be displayed. In how many ways can 3
identical guitars, 2 identical keyboards, and 3 identical violins be displayed?
18. A person walks from P to Q always traveling east or south. How many paths are possible?
•
P
•
Q
19. Suppose the letters of VERMONT are used to form “words.”
a) How many 7-letter “words” can be formed?
b) How many 6-letter “words” can be formed?
c) How many 5-letter “words” begin with a vowel and end with a consonant?
20. Telephone numbers in the U.S. and Canada have 10 digits as follows:
*3-digit area code number: first digit is not 0 or 1; second digit must be 0 or 1.
*3-digit exchange number: first and second digits are not 0 or 1
*4-digit line number: not all zeros
a) How many possible area codes are there?
b) The area code for downtown Chicago is 312. Within this area code how many exchange numbers are possible?
c) One of the exchange numbers for Chicago is 472. Within this exchange, how many line numbers are possible?
d) How many 7-digit phone numbers are possible in the 312 area code?
e) How many 10-digit phone numbers are possible in the U.S. and Canada?
60
Pre-Calculus
Combinatorics Days 1 & 2 Review
Name: ____________________________________
1. Jordan asked 142 students whether they like the shows “Storage Wars” and “Pawn Stars”. She found that 93 like “Storage Wars”,
49 like “Pawn Stars”, and 31 like both. How many people in Jordan’s survey:
a) do not like either show?
b) like only “Storage Wars”?
c) like only “Pawn Stars”?
2. A survey of 150 students at Oswego East High School showed the following: 52 play basketball, 74 play soccer, 31 play volleyball,
14 play basketball and soccer, 12 play basketball and volleyball, 14 play soccer and volleyball, and 5 play all three. How many
students:
a) Play soccer but neither of the other two?
b) Play basketball and soccer, but not volleyball?
c) Play none of the three?
d) Play just one of the three?
e) Play exactly two of the three?
f) Do not play volleyball?
3. Use the Venn diagram below to answer the questions that follow:
a) How many total people are represented in the diagram?
Coke
Pepsi
13
12
33
b) How many people like Sprite?
7
6
1
c) How many people like Sprite and Pepsi?
d) How many people like Sprite or Pepsi?
21
Sprite
14
e) How many people like only Coke?
4. You are eating dinner at a restaurant that offers 6 appetizers, 12 main dishes, 6 side orders, and 8 desserts. If you order one of each
of these, how many different dinners can you order?
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5. 8 swimmers are competing in the final round of a triathlon.
a) In how many different ways can the swimmers finish the competition?
b) In how many different ways can 4 of the swimmers finish in 1st, 2nd, 3rd, and 4th place?
For numbers 6 – 8 find the number of distinguishable permutations of the letters in the following words:
6. POP
7. KANSAS
8. THURSDAY
9. How many circular permutations are possible when seating five people around a table?
10. The window of a music store has 8 stands in fixed positions where instruments can be displayed. In how many ways can 3
identical guitars, 2 identical keyboards, and 3 identical violins be displayed?
11. A person walks from P to Q always traveling east or south. How many paths are possible?
•
P
•
Q
12. A person walks from P to Q always traveling east or south. How many paths are possible?
P
•
•
Q
13. Determine how many different 7-digit phone numbers are possible if (a) digits can be repeated, and (b) digits cannot be repeated
given the following conditions: Begins with 7 and ends with an even number.
14. Determine how many different 4-letter license plates are possible if (a) letters can be repeated, and (b) letters cannot be repeated
given the following conditions: Begins with a Q and ends with a vowel (Y is not a vowel).
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Combinatorics Day 3 Notes: Combinations
Combination: a selection of r objects from a group on n objects where the order is not important (or does not matter).
n
Cr =
n!
r!(n − r )!
Example 1:
a) Find the number of ways to purchase 3 different kinds of juice from a selection of 10 different juices.
b) Find the number of ways to rent 4 comedy DVDs from a collection of 9 comedy DVDs.
c) Find the number of ways 3 students can be selected from a committee of 5
Example 2: Deck of cards: when dealt a hand, the order you receive the cards does not matter.
a) Using a standard deck of 52 cards, how many different 7-card hands are possible?
b) How many of these hands have all 7 cards of the same suit?
c) How many of these 7 card hands have 4 kings and 3 other cards?
d) How many of these 7 card hands have 2 jacks, 3 sevens, and 2 aces?
e) How many possible 5 card hands contain exactly 3 kings?
Multiply / Add / or Subtract???
*When you are finding the number of ways both event A and event B can occur… MULTIPLY
*When you are finding the number of ways that event A or event B can occur…. ADD
*“At least” or “at most” …sometimes easier to SUBTRACT
(subtract the possibilities you do not want from the total number of possibilities)
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Example 3: A pizza parlor offers a selection of 3 different cheeses and 9 different toppings. In how many ways can a pizza be made
with the following:
a) 1 cheese and 2 toppings
b) 2 cheeses and 4 toppings
c) 3 cheeses and 1 topping
d) 2 cheeses or 3 toppings
Example 4:
a) You are taking a vacation and can visit as many as 5 different cities and 7 different attractions. Suppose you want to visit exactly 3
different cities and 4 different attractions. How many different trips are possible?
b) Suppose you want to visit at least 8 locations (cities or attractions). How many different types of trips are possible?
Example 5: From a group of 20 volunteers, you are choosing at least 18 to be peer counselors. In how many different ways can this be
done?
b) How many different ways can you choose at least 4 to be peer counselors?
Example 6: A restaurant offers 6 salad toppings. On a deluxe salad, you can have up to 4 toppings. How many different combinations
of toppings can you have?
Permutation or Combination??
*Permutations: ORDER MATTERS
*Combinations: ORDER DOES NOT MATTER
Example 7: Decide if the situation is a permutation or combination.
a) 4 recipes were selected for publication and 302 recipes were submitted.
b) 4 out of 200 contestants were awarded prizes of $100, $75, $50, and $25.
c) A president and vice-president are elected for a class of 210 students.
d) The batting order for the 9 starting players is announced.
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Pre-Calculus
Combinatorics Day 3 Homework
Name: _____________________________________
For numbers 1 – 4, find the number of combinations.
1. 10C6
2. 11C5
3. 14C14
4. 12C8
For numbers 5 – 9, find the number of possible 3-card hands that contain the cards specified.
5. 3 red cards
6. 3 aces
7. 3 face cards
8. 3 hearts
9. 3 of one kind (kings, queen, and so on)
For numbers 10 – 15, find the number of possible 5-card hands that contain the specified cards.
10. 4 aces and 1 king
11. 3 of one kind and 2 of a different kind
12. 3 face cards of the same suit and 2 other cards (none of which are face cards)
13. 5 black cards
14. 5 cards of the same suit
15. 5 cards, none of which are face cards (either kings, queens, or jacks)
16. A high school football team has 2 centers, 9 linemen (who can play either guard or tackle), 2 quarterbacks, 5 halfbacks, 5 ends,
and 6 fullbacks. The coach uses 1 center, 4 linemen, 2 ends, 2 halfbacks, 1 quarterback, and 1 fullback to form an offensive unit. In
how many ways can the offensive unit be selected?
17. An ice cream shop has a choice of 10 toppings. Suppose you can afford at most four toppings. How many different types of ice
cream sundaes can you order?
18. How many different rectangles occur in the grid shown below? (Hint: A rectangle is formed by choosing two of the vertical lines
in the grid and two of the horizontal lines in the grid.)
19. A pizza shop offers twelve different toppings. How many different three-topping pizzas can be formed with the twelve toppings?
(Assume no topping is used twice)
20. Nine people in your class want to be on a 5-person bowling team to represent the class. How many different teams can be chosen?
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66
Pre-Calculus
Combinatorics Days 1 – 3 Review Problems
Name: _____________________________________
1. How many 6 card hands from a standard 52 card deck contain exactly 4 red cards?
2. How many 6 card hands from a standard 52 card deck contain 3 aces, 2 kings and any other card?
3. How many 6 card hands from a standard 52 card deck contain 3 of one kind, and any other 3 cards?
4. How many 6 card hands from a standard 52 card deck contain 5 cards of the same suit and any other card of a different suit?
5. How many 6 card hands from a standard 52 card deck contain 3 hearts and 3 spades?
6. How many 6 card hands from a standard 52 card deck contain 4 faces cards and 2 other cards (neither of which are face cards)?
7. How many 6 card hands from a standard 52 card deck contain 3 face cards of the same suit and 3 other cards (none of which are
face cards)?
8. A survey of 200 people at a movie theater showed the following: 78 saw Safe Haven, 48 saw Die Hard, 84 saw Warm Bodies, 22
saw Safe Haven and Die Hard, 18 saw Safe Haven and Warm Bodies, 21 saw Die Hard and Warm Bodies, and 12 saw all three. How
many people saw:
a) Die Hard but neither of the other two?
b) Safe Haven and Die Hard, but not Warm Bodies?
c) none of the three?
d) just one of the three?
e) exactly two of the three?
f) did not see Warm Bodies?
9. Determine how many different 5-digit phone numbers are possible if (a) digits can be repeated, and (b) digits cannot be repeated
given the following conditions: Begins with a 3 or 5 and ends with an even number.
10. Nine swimmers are competing in the final round of a triathlon.
a) In how many different ways can the swimmers finish the competition?
b) In how many different ways can 3 of the swimmers finish in 1st, 2nd, and 3rd place?
11. You are eating dinner at a restaurant that offers 4 appetizers, 10 main dishes, 5 side orders, and 4 desserts. If you order one of
each of these, how many different dinners can you order?
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68
Combinatorics Day 4 Notes: The Binomial Theorem
Binomial Theorem
The binomial expansion of (a + b)n for any positive integer n is:
n
(a + b)n = nC0anb0 + nC1an – 1b1 + nC2an – 2b2 + ….. + nCna0bn = ∑ n C r a n−r b r
r =0
If you arrange the values of nCr in a triangular pattern in which each row corresponds to a value of n, you get Pascal’s Triangle:
0C0
1C0
2C0
1C1
2C1
2C2
Complete Pascal's Triangle by finding the underlying rule for generating the new values in each row.
Example 1: Expand the binomials.
a) (x + 4)3
b) (x – 5)5
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c) (2x + 3)4
d) (2x – y2)3
Example 2: Find the coefficient in the expression.
a) Find the coefficient of x2 in (3x + y)4
b) Find the coefficient of x in (x + 2)4
c) Find the coefficient of x7 in (2 – 3x)10
d) Find the 5th coefficient: (2x + y)5
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Pre-Calculus
Combinatorics Day 4 Worksheet
Name: _____________________________________
For numbers 1 and 2, expand the power of the binomial.
1. (2x + 1)5
2. (x3 + y)7
3. Find the coefficient of x7 in the expansion of (2x + 5)12.
4. Find the 6th coefficient: (x + 7y)10
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72
Pre-Calculus
Combinatorics Day 4 Homework
Name: _____________________________________
For numbers 1 – 4, expand the power of the binomial.
1. (x + 3)4
2. (x + y)3
3. (2x – y2)6
4. (4x + y3)4
5. Find the coefficient of x4 in the expansion of (2x + 1)7.
6. Find the coefficient of x5 in the expansion of (x + 3y)10.
7. Find the coefficient of x6 in the expansion of (4x – 3)10.
8. Find the 4th coefficient: (3x – 4y)7
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74
Pre-Calculus
Combinatorics Multiple Choice Packet
Name: _____________________________________
Day 1:
Use the following information to answer questions 1 – 8:
1,000 people are surveyed about whether they like Pepsi or Monster. Let P = the set of Pepsi drinkers, and let M = the set of Monster
drinker. These sets are shown in the Venn diagram below.
U
P
M
200
130
640
1. How many people do not like either Pepsi or Monster?
a) 30
b) 160
c) 200
d) 640
e) 970
b) 200
c) 330
d) 770
e) 970
b) 200
c) 330
d) 770
e) 970
b) 230
c) 670
d) 770
e) 840
b) 230
c) 670
d) 770
e) 840
b) 230
c) 640
d) 670
e) 870
b) 230
c) 640
d) 670
e) 870
b) 230
c) 640
d) 670
e) 870
d) 720
e) 1000
2. Find P ∩ M
a) 130
3. P ∪ M
a) 130
4. P ′
a) 200
5. M ′
a) 200
6. P ′ ∩ M
a) 200
7. P ∩ M ′
a) 200
8. P ′ ∪ M ′
a) 200
Day 2:
9. In how many ways can 10 runners finish a race first, second, or third?
a) 3
b) 10
c) 300
10. In an activity club with 30 students, the offices of president, vice president, and treasurer will be filled. In how many ways can the
offices be filled?
a) 30
b) 90
c) 150
d) 24,360
e) 27,000
11. How many different license plates are possible if two digits are followed by three letters?
a) 98
b) 876,450
c) 1,404,000
d) 1,757,600
e) 12,647,200
d) 360
e) 720
12. How many distinguishable permutations of the letters BANANA are there?
a) 60
b) 120
c) 180
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13. How many distinguishable permutations of the letters BALLOON are there?
a) 210
b) 340
c) 1260
d) 2880
e) 5040
14. How many different license plates are possible if three digits are followed by two letters?
a) 1560
b) 676,000
c) 1,320,000
d) 1,845,300
e) 2,225,200
d) 336
e) 512
d) 22,100
e) 132,600
15. In how many ways can 8 runners finish a race first, second, or third?
a) 3
b) 8
c) 24
Day 3:
16. In how many ways can 3 cards be chosen from a standard deck of 52 cards?
a) 156
b) 2210
c) 12,600
17. In how many ways can a 7 person committee be chosen from a group of 10 people?
a) 120
b) 1260
c) 12,600
d) 60,480
e) 604,800
18. A movie theater has 14 different movies showing. If you want to attend no more than 3 of the movies, how many different
combinations can you attend?
a) 142
b) 364
c) 470
d) 15,915
e) 16,387
d) 32,872
e) 212,961
19. In how many ways can 50 cards be chosen from a standard deck of 52 cards?
a) 1326
b) 6540
c) 16,445
20. An amusement park has 27 different rides. If you have 21 ride tickets, how many different combinations of rides can you take?
a) 567
b) 2320
c) 6740
d) 112,480
e) 296,010
d) 945
e) 2835
d) 360
e) 720
d) 811,008
e) 3,784,704
d) 20
e) 29
d) –2, 0, 4, 12, 28, …
e) 4, 9, 14, 19, 24, …
Day 4:
21. What is the coefficient of x4 in the expansion of (3x + 1)7?
a) 35
b) 105
c) 315
22. What is the coefficient of x3 in the expansion of (2x – 3)5?
a) –360
b) –120
c) 80
23. What is the coefficient of x8 in the expansion of (x + 4)12?
a) 1056
b) 14,080
c) 126,720
Sequences and Series Review:
24. What is the tenth term of the sequence defined by a n = 3n − 1 ?
a) 2
b) 8
c) 14
25. Which of the following is an arithmetic sequence?
a) 4, 8, 10, 17, 22, …
b) 2, 7, 9, 10, 15, …
c) 3, 9, 27, 81, 243, …
26. What is a rule for the nth term of the arithmetic sequence with a7 = 44 and a21 = 128?
a) a n = 6n − 2
b) a n = 2n − 6
c) a n = 2n + 6
d) a n = 6n + 2
e) a n = 6n + 11
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∑ (k
)
4
27. Which series is represented by
2
+k ?
k =2
a) 3 + 10 + 21
b) 10 + 21 + 36
c) 6 + 12 + 20
d) 10 + 14 + 18
e) 10 + 21 + 32
28. What is a rule for the nth term of the geometric sequence 3, –6, 12, –24, 48, …?
a) a n = 2(− 3)
n −1
b) a n = −3(2 )
c) a n = 3(− 2 )
n −1
n −1
d) a n = −3(− 2 )
n −1
e) a n = −2(3)
n −1
29. What is the sum of the first 10 terms of the geometric series 1 + 2 + 4 + 8 + 16 + …?
a) 240
b) 880
∞
30. What is the sum of the series:
∑ 3(0.4)
i −1
c) 1023
d) 1850
e) 4095
c) 4.25
d) 4.75
e) 5
?
i =1
a) 3.25
b) 3.75
31. What is the common ratio of an infinite geometric series whose sum is 625 and the first term is a1 = 125?
a) –8
b) –2
c)
4
5
d) 2
e) 8
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78
Pre-Calculus
Combinatorics Unit Review Sheet
Name: _____________________________________
Use the following information to answer questions 1 – 8:
800 people are surveyed about whether they have an XBox Kinect or a Nintendo Wii. Let X = the set of XBox Kinect owners, and let
N = the set of Nintendo Wii owners. These sets are shown in the Venn diagram below.
U
X
N
75
22
421
1. How many people do not have either an XBox Kinect or Nintendo Wii?
a) 22
b) 75
c) 282
d) 421
e) 518
b) 75
c) 282
d) 421
e) 518
b) 75
c) 282
d) 421
e) 518
b) 97
c) 282
d) 357
e) 703
b) 97
c) 282
d) 357
e) 703
b) 304
c) 357
d) 421
e) 703
b) 304
c) 357
d) 421
e) 778
b) 304
c) 357
d) 421
e) 778
2. Find X ∩ N
a) 22
3. Find X ∪ N
a) 22
4. Find X ′
a) 75
5. Find N ′
a) 75
6. Find X ′ ∩ N
a) 75
7. Find X ∩ N ′
a) 75
8. Find X ′ ∪ N ′
a) 75
9. Georgia asked 94 students whether they like the TV shows “Modern Family” and “Cougar Town”. She found that 57 like “Modern
Family”, 36 like “Cougar Town”, and 24 like both. How many people in Georgia’s survey:
a) do not like either show?
b) like only “Modern Family”?
c) like only “Cougar Town”?
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10. A survey of 100 students at New England College showed the following: 48 take English, 49 take history, 38 take language, 17
take English and history, 15 take English and language, 18 take history and language, and 7 take all three. How many students:
a) take history but neither of the other two?
b) take English and history, but not language?
c) take none of the three?
d) take just one of the three?
e) take exactly two of the three?
f) do not take language?
11. Use the Venn diagram below to answer the questions that follow:
a) How many total people are represented in the diagram?
Rock
11
18
b) How many people like country?
c) How many people like country and rap?
Rap
24
7
8
1
17
Country
14
d) How many people like country or rap?
e) How many people like at least two of the types of music?
For numbers 12 and 13, each event can occur in the given number of ways. Find the number of ways all of the events can occur.
12. Event 1: 2 ways, Event 2: 4 ways, Event 3: 5 ways
13. Event 1: 4 ways, Event 2: 6 ways, Event 3: 9 ways, Event 4: 7 ways
Use the following information for numbers 14 and 15: For the given configuration, determine how many different license plates are
possible if (a) digits and letters can be repeated, and (b) digits and letters cannot be repeated.
14. 3 letters followed by 3 digits.
15. 2 digits followed by 4 letters.
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16. a) How many 3-digit numbers contain no 7’s?
b) How many 3-digit numbers contain at least one 7?
17. How many different 5-digit zip codes are there if any of the digits 0 – 9 can be used?
18. You are going to set up a stereo system by purchasing separate components. In your price range you find 5 different receivers, 8
different CD players, and 12 different speaker systems. If you want one of each of these components, how many different stereo
systems are possible?
19. A pizza shop runs a special where you can buy a large pizza with one cheese, one vegetable, and one meat for $9.00. You have a
choice of 7 cheeses, 11 vegetables, and 6 meats. Additionally, you have a choice of 3 crusts and 2 sauces. How many different
variations of the special pizza are possible?
20. To keep computer files secure, many programs require the user to enter a password. The shortest allowable passwords are
typically six characters long and contain both letters and numbers. How many six-character passwords are possible if (a) characters
can be repeated and (b) characters cannot be repeated.
21. Evaluate :
10!
5!2!
22. How many different ways can 4 friends stand in a cafeteria line?
For numbers 23 – 26, find the number of distinguishable permutations of the letters in the word.
23. JET
24. DAD
25. ALABAMA
26. MISSISSIPPI
27. How many different ways can the 15 member student council committee vote on a president and a vice-president?
28. “Ringing the changes” is a process where the bells in a tower are rung in all possible permutations. Westminster Abbey has 10
bells in its tower. In how many ways can its bells be rung?
In numbers 29 – 31, find the number of possible 4-card hands that contain the cards specified.
29. 4 red cards
30. 4 aces
31. 4 face cards
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For numbers 32 – 36, find the number of possible 5-card hands that contain the cards specified.
32. 5 face cards
33. 4 aces and 1 other card
34. 1 ace and 4 other cards
35. 2 aces and 3 kings
36. 4 of one kind (kings, queen, and so on) and 1 of a different kind
For numbers 37 – 40, use the binomial theorem to write the binomial expansion.
37. (x + 4)3
38. (x – 5)5
39. (x3 + 3)5
40. (2x – y2)7
41. Find the coefficient of x5 in the expansion of (x – 3)7.
42. Find the coefficient of x4 in the expansion of (x + 2)8.
43. Find the coefficient of x8 in the expansion of (3x – 2)10
44. Find the coefficient of x2 in the expansion of (x – 4y)6
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Probability
83
84
Probability Day 1 Notes: Introduction to Probability
Probability: the likelihood that the event will occur.
*must be a number between 0 and 1.
If an event is certain to occur, it has a probability of 1.
If an event cannot occur, it has a probability of 0.
1
.
2
If an event is equally likely to occur or not occur, it has a probability of
Theoretical Probability
When all outcomes are equally likely that an event will occur is: P(A) =
number of outcomes in A
total number of outcomes
Simply called the probability of an event.
Example 1: A spinner has 8 equal-size sectors numbered from 1 to 8. Find the probability:
a) of spinning a 6.
b) of spinning an even number.
c) of spinning a number greater than 5.
Example 2: One card is drawn from a standard 52-card deck. Find the probability:
a) of choosing a red card.
b) of choosing an even numbered card.
c) of choosing a face card.
d) of choosing a diamond.
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Example 3: You roll a six-sided die once. Find the probability of:
a) rolling a 6.
b) rolling an even number.
c) rolling a prime number.
d) rolling a factor of 10.
e) rolling a composite number.
f) rolling a perfect square.
Experimental Probability
This is used when it is impossible or inconvenient to find the theoretical probability. It can be done by performing an experiment,
conducting a survey, or looking at the history of the event.
Example 4: Ninth graders must enroll in one math class. The enrollments of ninth grade students during the previous year are shown
in the bar graph. Find the probability that a randomly chosen student from this year’s 9th grade class is enrolled in
a) Consumer Math.
100
87
80
69
60
51
36
40
b) Algebra 1 or Intro to Algebra.
20
0
Algebra 1
Consum er
Math
Geom etry
Intro to
Algebra
Example 5: You made 15 of 21 free throw attempts. Find the probability that you will make your next free throw.
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Pre-Calculus
Probability Day 1 Homework
Name: _____________________________________
For numbers 1 – 8, you have an equally likely chance of choosing any integer from 1 through 50. Find the probability of the given
event.
1. An even number is chosen.
2. A number less than 35 is chosen.
3. A perfect square is chosen.
4. A prime number is chosen.
5. A factor of 150 is chosen.
6. A multiple of 4 is chosen.
7. A two-digit number is chosen.
8. A perfect cube is chosen.
For numbers 9 – 14, a card is randomly drawn from a standard deck of 52 cards. Find the probability of drawing the given card.
9. The king of diamonds.
10. A king.
11. A spade.
12. A black card.
13. A card other than a 2.
14. A face card (a king, queen, or jack)
For numbers 15 – 18, you randomly choose a marble from a bag. The bag contains 10 black, 8 red, 4 white, and 6 blue marbles. Find
the probability of the following:
15. Choosing a white marble.
16. Choosing a blue marble.
17. Choosing a red marble.
18. Choosing a black marble.
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88
Probability Day 2 Notes: Probability of Compound Events
Union: when you consider ALL the outcomes for either of two events A and B. (A, B, A & B)
Intersection: when you consider only the outcomes shared by both A and B. (A & B)
Mutually Exclusive Events: if there is no intersection of A & B (Nothing in common)
IF A & B INTERSECT:
P(A or B) = P(A) + P(B) – P(A & B)
(Since P(A) and P(B) both include P(A & B))
IF A & B ARE MUTUALLY EXCLUSIVE:
P(A or B) = P(A) + P(B)
(do not intersect; have nothing in common, disjoint)
A




B



A


UNION of A and B
P(A or B)




B



A



INTERSECTION of A and B
P(A and B)







B

INTERSECTION is empty
mutually exclusive events
Example 1: Find the indicated probability.
a) P(A) = 0.4, P(B) = 0.45
1
5
b) P(A) = , P(B) =
9
9
2
9
P(A and B) = 0.1
P(A or B) =
P(A or B) = _________
P(A and B) = ________
Example 2: One six-sided die is rolled.
a) What is the probability of rolling a 5 or a multiple of 3?
b) What is the probability of rolling a multiple of 3 or a multiple of 2?
Example 3: A card is randomly selected from a standard deck of 52 cards.
a) What is the probability that it is a 10 or a face card?
b) What is the probability that it is a face card or a spade?
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Example 4: In a poll of high school juniors, 6 out of 15 took a French class and 11 out of 15 took a math class. Fourteen out of 15
took French or math. What is the probability that a student took both French and Math?
Example 5: Out of 200 students in a senior class, 113 students are either varsity athletes or on the honor roll. There are 74 varsity
athletes and 51 seniors who are on honor roll. What is the probability that a randomly selected senior is both a varsity athlete and on
the honor roll?
Example 6: In a survey of 200 pet owners, 103 owned dogs, 88 owned cats, 25 owned birds, and 18 owned reptiles.
a) None of the respondents owned both a cat and a bird. What is the probability that they owned a cat or a bird?
b) Of the respondents, 52 owned both a cat and a dog. What is the probability that a respondent owned a cat or a dog?
c) Of the respondents, 119 owned a dog or a reptile. What is the probability that they owned a dog and a reptile?
Using Complements to Find Probability
Complement of event A: all outcomes that are NOT in A.
Denoted A′ (A prime)
**The probability of the complement of A is P(A′) = 1 – P(A)
Example 7: A card is randomly selected from a standard deck of 52 cards. Find the probability of the given event:
a) The card is not a king.
b) The card is not an ace or a jack.
Example 8: When two six-sided dice are rolled, there are 36 possible outcomes. Find the probability of the given event:
a) The sum is not six.
b) The sum is less than or equal to 9.
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Pre-Calculus
Probability Day 2 Worksheet
Name: _____________________________________
For numbers 1 and 2, events A and B are disjoint. Find P(A or B)
2
1
1. P(A) = , P(B) =
3
5
2. P(A) = 0.55, P(B) = 0.2
For numbers 3 and 4, find the indicated probability.
3. P(A) = 0.6, P(B) = 0.2
P(A or B) = 0.7
P(A and B) = ________
6
3
, P(B) =
11
11
7
P(A or B) =
11
P(A and B) = __________
4. P(A) =
For numbers 5 and 6, find P(A′)
5. P(A) = 0
6. P(A) =
5
8
For numbers 7 – 9, a card is randomly selected from a standard deck of 52 cards. Find the probability of drawing the given card.
7. A king or a diamond
8. A 4 or a 5
9. Not a heart
For numbers 10 and 11, find the indicated probability. State whether A and B are disjoint (mutually exclusive) events.
10. P(A) = 0.6, P(B) = 0.32
P(A or B) = _________
P(A and B) = 0.25
1
1
11. P(A) = , P(B) =
6
2
2
P(A or B) =
3
P(A and B) = __________
For numbers 12 and 13, two six-sided dice are rolled. Find the probability of the given event.
12. The sum is not 7.
13. The sum is less than 8 or greater than 11.
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92
Pre-Calculus
Probability Day 2 Homework
Name: _____________________________________
For numbers 1 and 2, events A and B are disjoint. Find P(A or B)
2
3
2. P(A) = , P(B) =
5
5
1. P(A) = 0.3, P(B) = 0.1
For numbers 3 and 4, find the indicated probability.
2
4
4. P(A) = , P(B) =
7
7
3. P(A) = 0.5, P(B) = 0.35
1
7
P(A and B) = 0.2
P(A and B) =
P(A or B) = _________
P(A or B) = ________
5. A twelve-sided die is rolled. (The sides are numbered 1 – 12). What is the probability of rolling a multiple of 2 or a multiple of 5?
6. In a poll of adults, 284 out of 677 own an IPAD and 251 own a Kindle Fire. Four hundred and sixty five out of 677 own either an
IPAD or a Kindle Fire. What is the probability that an adult from this poll owns both an IPAD and a Kindle Fire?
For numbers 7 – 9, a card is randomly selected from a standard deck of 52 cards. Find the probability of drawing the given card.
7. A king and a diamond
8. A spade or a club
9. A 6 and a face card
For numbers 10 and 11, find the indicated probability. State whether A and B are disjoint (mutually exclusive) events.
10. P(A) = 0.25, P(B) = 0.4
P(A or B) = 0.5
P(A and B) = ___________
8
, P(B) = ______
15
3
P(A or B) =
5
2
P(A and B) =
15
11. P(A) =
93
For numbers 12 and 13, two six-sided dice are rolled. Find the probability of the given event.
12. The sum is 3 or 4.
13. The sum is greater than or equal to 5.
14. You and your best friend are among several candidates running for class president. You estimate that there is a 45% chance you
will win and a 25% chance your best friend will win. What is the probability that either you or your best friend win the election?
For numbers 15 and 16, find P(A′)
15. P(A) = 0.5
16. P(A) =
1
3
Use the following information for numbers 17 and 18: In a survey of 100 radio listeners, 44 like rock, 43 like rap, and 33 like country.
17. Of the respondents, 18 like rock and rap. What is the probability that a listener likes rock or rap?
18. Of the respondents, 68 like country or rap. What is the probability that they likes country and rap?
19. A Pre-Calculus class has 32 students. 17 are wearing red shirts and 24 are wearing jeans. Eleven are wearing both red shirts and
jeans. Find the probability that a randomly chosen student is wearing a red shirt or jeans?
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Probability Day 3 Notes: Probability of Independent Events
Independent Events
If the occurrence of one event has NO effect on the occurrence of the other.
IF A & B ARE INDEPENDENT:
P(A & B) = P(A) ⋅ P(B)
Example 1: For a fundraiser, a class sells 150 raffle tickets for a mall gift certificate and 200 raffle tickets for a booklet of movie
passes. You buy 5 raffle tickets for each prize. What is the probability that you win both prizes?
Example 2: In a BMX meet, each heat consists of 8 competitors who are randomly assigned lanes from 1 to 8. What is the
probability that a racer will draw lane 8 in the 3 heats in which the racer participates?
Dependent Events: If the occurrence of one event affects the occurrence of the other.
Conditional Probability: the probability that B will occur given that A has occurred.
P(B | A)  “Probability of B occurring given that A occurred ”
IF A & B ARE DEPENDENT:
P(A & B) = P(A) ⋅ P(B | A)
Finding Conditional Probability:
Example 3: Let n be a randomly selected integer from 20 to 40. Find the indicated probability.
a) n is even given that it is composite
b) n is 34 given that it is greater than 30
Example 4: You randomly select two cards from a standard 52-card deck. Find the probability that the first card is a diamond and the
second card is red….
a) if you replace the first card before selecting the 2nd card.
b) if you do not replace the first card.
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Three or More Dependent Events
P(A and B and C) = P(A) ⋅ P(B | A) ⋅ P(C | A and B)
Example 5: You randomly select three cards from a standard 52-card deck without replacement. Find the probability that the first
card is a ten, the second card is an ace, and the third card is a face card.
Example 6: You randomly choose 4 marbles from a bag without replacement. The bag contains 10 black, 8 red, 4 white, and 6 blue
marbles. Find the probability of pulling:
a) a black, then a red, then a black, then a white.
b) a red, then a white, then a blue, then a red.
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Pre-Calculus
Probability Day 3 Worksheet
Name: _____________________________________
For numbers 1 and 2, events A and B are independent. Find the indicated probability.
1. P(A) = 0.3
P(B) = 0.4
P(A and B) = __________
2. P(A) =_______
P(A and B) = 0.6
P(B) = 0.8
For numbers 3 – 6, you are playing a game that involves spinning the wheel shown. Find the probability of spinning the given colors.
5 red pieces
3. blue, then yellow
4. red, then green
4 yellow pieces
4 green pieces
3 blue pieces
5. red, then blue
6. red, then yellow, then red
For numbers 7 and 8, events A and B are dependent. Find the indicated probability.
7. P(A) = 0.7
P(B | A) = 0.5
P(A and B) = __________
8. P(A) =_______
P(B | A) = 0.4
P(A and B) = 0.2
For numbers 9 and 10, let n be a randomly selected integer from 1 to 20. Find the indicated probability.
9. n is odd given that it is prime
10. n is 5 given that it is less than 8
For numbers 11 and 12, find the probability of drawing the given cards from a standard deck of 52 cards (a) with replacement and (b)
without replacement.
11. A 10, then a 2
12. A queen, then an ace
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98
Pre-Calculus
Probability Day 3 Homework
Name: _____________________________________
For numbers 1 and 2, events A and B are independent. Find the indicated probability.
1. P(A) = 0.4
P(B) = 0.6
P(A and B) = _________
2. P(A) = 0.5
P(B) =_______
P(A and B) = 0.1
For numbers 3 – 6, you are playing a game that involves spinning the wheel shown. Find the probability of spinning the given colors.
3. green, then blue
4. red, then yellow
5. blue, then red
6. blue, then green, then red
5 red pieces
4 yellow pieces
4 green pieces
3 blue pieces
For numbers 7 and 8, events A and B are dependent. Find the indicated probability.
7. P(A) = 0.3
P(B | A) = 0.6
P(A and B) = _________
8. P(A) = 0.8
P(B | A) =_______
P(A and B) = 0.32
For numbers 9 and 10, let n be a randomly selected integer from 1 to 20. Find the indicated probability.
9. n is 2 given that it is even
10. n is prime given that it has 2 digits
For numbers 11 and 12, find the probability of drawing the given cards from a standard deck of 52 cards (a) with replacement and (b)
without replacement.
11. A club, then a spade
12. A face card, then a 6
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100
Pre-Calculus
Quiz Review Probability Days 1 – 3
Name: _____________________________________
For numbers 1 – 8, you have an equally likely chance of rolling any value on each of two dice. Find the probability of the given event.
1. rolling a sum of either 7 or 9
2. rolling a sum greater than 5
3. rolling a 6 on exactly one die
4. rolling doubles
5. The sum is even and a multiple of 3.
6. The sum is not 2 or 12.
7. The sum is greater than 7 or odd.
8. The sum is prime and even.
Use the following information for numbers 9 – 11:
Your cousin lives on a small farm. She is a member of the 4-H Club and is showing nine animals at the county fair. Two of her
animals won a blue ribbon (1st place), one won a red ribbon (2nd place), and three won white ribbons (3rd place). You do not know
which animals won which prizes. You choose one of your cousin’s animals at random.
9. What is the probability that the animal won a 1st place ribbon?
10. What is the probability that the animal won a ribbon?
11. What is the probability that the animal won a red or white ribbon?
Use the following information for numbers 12 – 14:
Of all live births in the United States in 1996, 12.9% of the mothers were teenagers, 51.8% were in their twenties, 33.4% were in their
thirties, and the rest were in their forties. Suppose a mother is chosen at random.
12. What is the probability that the mother gave birth in her twenties?
13. What is the probability that the mother gave birth in her twenties or thirties?
14. What is the probability that the mother gave birth in her forties?
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15. Students at Northwestern High School have three choices for a required science in their junior year: physics, chemistry, or biology.
Experience has shown that the probability of a student selecting physics is 0.12 and the probability of a student selecting chemistry is
0.57. If each student can select only one science course, what is the probability that a randomly chosen student will select biology?
16. A parking lot has 25 cars. Eight are red and 13 have four doors. Six are both red and have four doors. Find the probability that a
randomly chosen car will be red or have four doors.
17. A card is randomly selected from a standard deck of 52 cards. What is the probability that it is a king or queen?
18. A card is randomly selected from a standard deck of 52 cards. What is the probability that it is a diamond or a 10?
For numbers 19 – 22, a jar contains 12 red marbles, 16 blue marbles, and 18 white marbles. Find the probability of choosing the given
marbles from the jar (a) with replacement and (b) without replacement.
19. red, then blue
20. white, then white
21. red, then white, then blue
22. red, then red, then white
23. The probability of selecting a rotten apple from a basket is 12%. What is the probability of selecting three good apples when
selecting one from each of three different baskets?
24. You randomly select two cards from a standard 52-card deck. What is the probability that the first card is a king or queen and the
second card is a king, queen, or jack if you replace the first card before selecting the second?
25. In number 24, what is the probability if you do not replace the first card before selecting the second?
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Probability Day 4 Notes: Permutations and Combinations
Remember:
*If the order is important, use a permutation.
*If the order is NOT important, use a combination.
Combinations:
Example 1: Five cards are drawn from a standard 52-card deck. Find the probability:
a) of choosing exactly one red card.
b) of choosing exactly one even numbered card.
c) of choosing exactly three black cards.
d) that exactly 2 cards are red.
Example 2: You put a picture CD that contains 7 pictures in your computer. You set the options to play the slideshow in a random
order. The computer plays all 7 pictures without repeating any. You have 3 favorite pictures on the disc. What is the probability that
2 of your favorites play first, in any order?
Example 3: Five coins are chosen at random from a purse containing 10 pennies and 6 nickels.
a) What is the probability that the selection will contain at least one penny?
b) What is the probability that the selection will contain at least one nickel?
c) What is the probability that the selection will only contain 3 pennies?
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Permutations:
Example 4: You put a picture CD that contains 7 pictures in your computer. You set the options to play the slideshow in a random
order. The computer plays all 7 pictures without repeating any. Find the probability that the pictures are played in the same order
they were put on the disc.
Example 5: You have 8 pennies in your pocket dated 1972, 1978, 1979, 1985, 1989, 1991, 1993, and 1999. You take the coins out of
your pocket one at a time. What is the probability that they are taken out in order?
Example 6: Three children have a choice of 12 summer camps that they can attend. If they each randomly choose which camp to
attend, what is the probability that they attend all different camps?
Example 7: A family of 4 is each choosing 1 of 8 possible vacations. What is the probability that each family member picks a
different vacation for his or her first choice?
Example 8: One high school requires students to complete 30 hours of community service to graduate. There are 156 different
community service options from which to choose. What is the probability that in a group of 5 students, at least 2 of them will be
doing the same community service?
Example 9: A restaurant gives a free fortune cookie to every guest. The restaurant claims there are 500 different messages hidden
inside the fortune cookies. What is the probability that a group of 5 people receive at least 2 fortune cookies with the same message
inside?
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Example 10: Seven different prizes are being given in a raffle contest. 157 tickets are sold. After each prize is called, the winning
ticket is returned to the drawing box and is eligible to be picked for another prize. What is the probability that at least one of the
tickets is drawn twice?
Can you Decide???
Example 11: While you are riding to school, your IPOD randomly plays 4 different songs from a play list that has 16 songs on it.
What is the probability that you will hear your favorite song?
Example 12: Tim has 4 table tennis balls with small cracks. His friend accidentally mixed them in with 16 good balls. If Tim
randomly picks 3 table tennis balls, what is the probability that at least 1 is cracked?
Example 13: Find the probability of winning the lottery according to the given rules. Assume numbers are selected at random.
a) You must correctly select 6 out of 48 numbers. The order of the numbers is not important.
b) You must correctly select 4 numbers, each an integer from 0 to 9. The order of the numbers is important.
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106
Pre-Calculus
Probability Day 4 Part 1 Homework
Name: _____________________________________
1. Three cards are drawn from a well-shuffled standard deck of 52 cards, one after the other and without replacement.
Find the probability of drawing:
a) all hearts
b) no hearts
c) at least one heart
2. Free concert tickets are distributed to 4 students chosen at random from 8 juniors and 12 seniors in the school orchestra. What is
the probability that free tickets are received by:
a) 4 seniors?
b) exactly 3 seniors?
d) exactly 1 senior?
e) no seniors?
c) exactly 2 seniors?
3. A town council consists of 8 Democrats, 7 Republicans, and 5 Independents. A committee of 3 is chosen by randomly pulling
names from a hat. What is the probability that the committee has:
a) 2 Democrats and 1 Republican?
b) 3 Independents?
c) no independents?
d) 1 Democrat, 1 Republican, and 1 Independent
4. While you are riding to school, your IPOD randomly plays 3 different songs from a play list that has 26 songs with each song title
starting with a different letter of the alphabet. What is the probability that you will hear the song titles that start with A, B, and C, in
that order?
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5. A committee of 3 people is to be randomly selected from 6 people Adam (A), Ben (B), Cathy (C), Denise (D), Eleanor (E), and
Frank (F).
a) Find the probability that Eleanor is on the committee.
b) Find the probability that Eleanor and Frank are on the committee.
c) Find the probability that Eleanor or Frank are on the committee.
d) Find the probability that neither Eleanor nor Frank is on the committee.
e) Find the probability that Adam is on the committee and Ben is not.
f) Find the probability that Adam and Ben are on the committee but Cathy is not.
6. The letters of the word COMBINE are put into a bag. If three letters are taken out and not replaced, what is the probability the three
letters are B, M, and C, in that order?
7. The following letters LVEWOS are put into a bag. If all the letters are taken out and not replaced, what is the probability that the
order you take them out spells WOLVES?
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Pre-Calculus
Day 4 Part 2 Homework
Name: _____________________________________
1. A community center hosts a talent contest for local musicians. On a given evening, seven musicians are scheduled to perform. The
order of which the musicians perform is randomly selected during the show. What is the probability that the musicians perform in
alphabetical order by their last names? (Assume that no two musicians have the same last name).
2. A cereal company plans to put five new cereals on the market: a wheat cereal, rice cereal, corn cereal, oat cereal, and a multi-grain
cereal. The order in which the cereals are introduced will be randomly selected. Each cereal will have a different price. What is the
probability that the cereals are introduced in order of their suggested retail price?
3. You create a video montage that contains 5 of your favorite YouTube clips. You set the option to play the videos in a random
order. The computer plays all 5 videos without repeating any. Find the probability that the videos are played in the same order they
were put on the disc.
4. You have 3 dimes in your pocket dated 1978, 1981, and 2013. You take the coins out of your pocket, one at a time. What is the
probability that they are taken out in that order?
5. Four children have a choice of 9 water parks that they can attend over summer vacation. If they each randomly choose which water
park to attend, what is the probability that they attend all different water parks?
6. A family of 5 is each choosing 1 of 14 possible restaurants. What is the probability that each family member picks a different
restaurant for his or her first choice?
7. One university requires students to complete 25 hours of community service to be accepted. There are 30 different community
service options on campus from which to choose. What is the probability that in a group of 5 students, at least 2 of them will be doing
the same community service?
8. The Easter Bunny gives a free Easter egg to every guest that comes to visit. The mall claims there are 102 different prizes hidden in
the Easter eggs. What is the probability that a group of 6 people receive at least 2 Easter eggs with the same prize inside?
9. Four letter “words” are formed from the letters A, B, C, D, E, F, and G. Find the probability that
a) the “word” contains no vowels.
b) the word formed is FADE.
10. Five prizes are being given in a raffle contest. Two hundred and seventy nine tickets are sold. After each ticket number is called,
the winning ticket is returned to the drawing box and is eligible to be picked for another prize. What is the probability that at least one
of the tickets is drawn twice?
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110
Pre-Calculus
Day 4 Part 3 Homework
Name: ____________________________________
Use the following information for numbers 1 – 4: Five marbles are chosen at random from a jar containing 12 red marbles and 8 blue
marbles.
1. What is the probability that the selection will contain at least one red marble?
2. What is the probability that the selection will contain at least one blue marble?
3. What is the probability that the selection will contain exactly one red marble?
4. What is the probability that the selection will contain exactly three blue marbles?
Use the following information for numbers 5 – 8: Six cards are drawn from a standard 52 card deck. Find the probability of the
following.
5. There are 3 kings and 3 other cards.
6. All the cards are red.
7. There are 2 aces and 4 other cards.
8. Two of the cards are even and four are odd.
9. a) The letters of the word STUDENT are put into a bag. If three letters are taken out and not replaced, what is the probability the
three letters are S, D, and E, in that order?
b) The following letters STODUN are put into a bag. If all the letters are taken out and not replaced, what is the probability that the
order you take them out spells DONUTS?
10. What is the probability that 5 books placed randomly on a shelf will be placed alphabetically?
11. There are 25 students in your class and your teacher is choosing 4 to help the secretary move books. If your teacher randomly
chooses 4 students, what is the probability that you and your 3 best friends will be selected (assuming he/she is not biased)?
111
12. Six boys and six girls belong to a club. Four officers are to be selected at random. What is the probability that they will all be
girls?
13. You put a CD that has 8 songs in your CD player. You set the player to play the songs at random. The player plays all 8 songs
without repeating any song. What is the probability that the songs are played in the same order they are listed on the CD?
14. There are 16 teams in the state basketball tournament. If each team has an equal chance of winning, what are the chances that the
places will go as follows:
1st place – West Jordan HS
2nd place – Riverton
3rd place – Bingham
4th place – Copper Hills
15. There are 204 students in the 10th grade. Five of these students will be selected randomly to represent your class on a 5-person
bowling team. What is the probability that the team chosen will be you and your 4 best friends?
16. What are all the different ways the letters HTAM can be arranged? What is the probability that if you randomly selected one of
these arrangements, you would select the one that spells MATH?
17. A local charity group is conducting a raffle where 50 tickets are to be sold – one per customer. There are three different prizes to
be awarded. If the four organizers of the raffle each buy one ticket, what is the probability that the organizers will win all of the
prizes?
18. Suppose that 5 people are chosen at random. Find the probability that at least two of the people share the same birthday.
19. The organizer of a potluck dinner sends 5 people a list of 8 different recipes and asks each person to bring one of the items on the
list. If all 5 people randomly choose a recipe from the list, what is the probability that at least 2 will bring the same thing?
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Probability Day 5 Notes: Binomial Distributions
Binomial Experiments
*There are n independent trials.
*Each Trial has only two possible outcomes: success and failure.
*The probability of success is the same for each trial. This probability is denoted by p. The probability of failure is given by 1− p.
We will be doing binomial probabilities using the calculator:
Keyword “exact”
…find probability with binompdf(number of trials, probability of success, number of successes)
Keywords “less than or equal to”
…find probability with binomcdf(number of trials, probability of success, number of successes)
Keywords “at least” or “greater than or equal to”
… find probability with 1 – binomcdf with 1 less number for success
What you are doing here is 1 minus what you DO NOT WANT and that will give you the probability that you do want.
Example 1:
a) Calculate the probability of tossing a coin 20 times and getting exactly 9 heads.
b) Calculate the probability of tossing a coin 20 times and getting less than 6 heads.
c) Calculate the probability of tossing a coin 32 times and getting at least 14 heads.
d) About 1% of people are allergic to bee stings. What is the probability that exactly 1 person in a class of 25 is allergic to bee stings?
e) Refer to (d), what is the probability that 4 or more of them are allergic to bee stings?
Example 2:
a) At a college, 53% of students receive financial aid. In a random group of 9 students, what is the probability that exactly 5 of them
receive financial aid?
b) Now using (a), what is the probability that fewer than 3 students in the class receive financial aid?
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Pre-Calculus
Probability Day 5 Homework
Name: _____________________________________
For numbers 1 – 4, calculate the probability of tossing a coin 25 times and getting the given number of heads.
1. 2
2. 10
3. 18
4. 25
For numbers 5 – 8, calculate the probability of randomly guessing the given number of correct answers on a 20-question multiple
choice exam that has choices A, B, C, and D for each question.
5. 10
6. 8
7. 18
8. 5
For numbers 9 – 12, calculate the probability of q successes for a binomial experiment consisting of n trials with probability p of
successes on each trial.
9. q ≥ 4, n = 8, p = 0.35
10. q ≤ 5, n = 10, p = 0.55
11. q ≤ 3, n = 5, p = 0.7
12. q ≥ 2, n = 5, p = 0.6
13. An automobile-safety researcher claims that 1 in 10 automobile accidents is caused by driver fatigue. What is the probability that
at least three of five automobile accidents are caused by fatigue?
14. The probability is 0.58 that a car stolen in a city in the United States will be returned to its lawful owner. Suppose that in one day
30 cars were stolen. What is the probability that at least 25 of these stolen cars will be returned to their lawful owners?
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116
Probability Day 6 Notes: Working with Conditional Probability and Expected Value
We are going to consider the probability of a certain cause when a certain effect is observed. For example, a flu can cause symptoms
such as high fever and sore throat, but these can be symptoms of other disorders besides flu. The first example examines the
probability that a person who has a fever (an effect) also has the flu (a possible cause of the fever).
Example 1: During a flu epidemic, 35% of a school’s students have the flu. Of those with the flu, 90% have high temperatures.
However, a high temperature is also possible for people without the flu; in fact, the school nurse estimates that 12% of those without
the flu have high temperatures.
a) Incorporate the facts given above into a tree diagram.
b) About what percent of the student body have a high temperature?
c) If a student has a high temperature, what is the probability that the student has the flu?
Example 2:
a) Find the missing probabilities in the tree diagram:
0.3
?
X′
X
0.6
?
?
Y
Y′
Y
0.8
Y′
b) Find P(X and Y)
c) Find P(X ′ and Y)
d) Find P(Y)
e) P(X | Y)
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Expected Value
A collection of outcomes is partitioned into n events, no two events which have any outcomes in common. The probabilities of the n
events occurring are p1, p2, p3, …, pn where p1 + p2 + p3 + … + pn = 1. The values of the n events are x1, x2, x3, …, xn. The expected
value V of the collection of outcomes is the sum of the products of the events’ probabilities and their values.
V = p1 x1 + p2 x2 + p3 x3 + … + pn xn
Any games or situations where you can win or lose points based on certain outcome rules, and involves probabilities for each, relates
to the concept of expected value.
Example 3: Find the expected value:
a)
Payoff
Probability
30
0.33
27
0.25
24
0.42
b) Payoff
Probability
10
0.4
–7
0.2
–15
0.4
Example 4: You and a friend each flip a coin. If both coins land heads up, then your friend scores 3 points and you lose 3 points. If
one or both of the coins land tails up, then you score 1 point and your friend loses 1 point. What is the expected value of the game
from YOUR point of view?
Example 5: A box contains 5 red balls and 4 green balls. Two balls are randomly chosen without replacement. If both are red, you
win $4. If just one is red, you win $2. Otherwise you lose $2. What is your expected gain or loss?
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Pre-Calculus
Probability Day 6 Homework
Name: _____________________________________
1. Using observations made of drivers arriving at a certain high school, a study reports that 69% of adults wear seat belts while
driving. A high school student also in the car wears a seat belt 66% of the time when the adult wears a seat belt, and 26% of the time
when the adult does not wear a seat belt. What is the probability that a high school student in the study wears a seat belt?
2. A high school basketball team leads at halftime in 60% of the games in a season. The team wins 80% of the time when they have
the halftime lead, but only 10% of the time when they do not. What is the probability that the team wins a particular game during the
season?
3. A tennis player wins a match 55% of the time when she serves first and 47% of the time when her opponent serves first. The player
who serves first is determined by a coin toss before the match. What is the probability that the player wins the given match?
119
4. Use the tree diagram to the right to find each probability.
a) P(X ′ )
0.4
b) P(Y ′ | X)
X′
X
0.3
c) P(Y ′ | X ′ )
d) P(X and Y)
e) P(X ′ and Y)
f) P(Y)
Y
0.5
Y′
Y
Y′
g) P(X | Y)
5. Find the expected value:
a)
Payoff
Probability
60
0.4
52
0.5
50
0.1
b) Payoff
Probability
13
0.4
–3
0.2
–12
0.4
6. A box contains 2 red balls and 1 white ball. Two balls are randomly chosen without replacement. If both are red, player A wins $5
from player B. Otherwise player B wins $2 from player A. Is this game fair?
7. Two dice are rolled. If the sum of the numbers showing on the dice is odd, player A wins $1 from player B. If both dice show the
same the number, player A wins $3 from player B. Otherwise player B wins $3 from player A. Is this game fair?
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Pre-Calculus
Probability Multiple Choice Packet
Name: _____________________________________
Day 1:
1. What is the probability of rolling an even number if you roll a six-sided die with sides numbered from 1 through 6?
a)
1
6
b)
1
4
c)
1
3
d)
1
2
e) 1
2. What is the probability of rolling a factor of 20 if you roll a six-sided die with sides numbered from 1 through 6?
a)
1
6
b)
1
3
c)
1
2
d)
2
3
e) 1
3. You have an equally likely chance of choosing any number from 1 to 15. What is the probability that you choose a number greater
than 10?
a)
1
15
b)
1
10
c)
1
5
d)
1
3
e)
4
5
4. You have an equally likely chance of choosing any number from 1 to 15. What is the probability that you choose a number less
than or equal to 12?
a)
1
15
b)
1
10
c)
1
5
d)
1
3
e)
4
5
5. You have an equally likely chance of choosing any number from 1 to 10. What is the probability that you choose a number less
than 4?
a)
3
10
b)
2
5
c)
1
2
d)
2
3
e)
4
5
6. One card is drawn from a standard 52 card deck, what is the probability that it is an odd numbered card?
a) 0.1923
b) 0.3077
c) 0.3846
d) 0.4615
e) 0.5
Day 2:
7. A card is randomly selected from a standard deck of 52 cards. What is the probability that it is a 10 or a diamond?
a) 0.285
b) 0.308
c) 0.405
d) 0.430
e) 0.504
8. A card is randomly selected from a standard deck of 52 cards. What is the probability that it is an 8 or a king?
a) 0.082
b) 0.106
c) 0.154
d) 0.208
e) 0.25
d) 0.58
e) 0.61
d) 0.57
e) 0.69
d) 0.58
e) 0.61
d) 0.6
e) 0.8
9. If P(A) = 0.4, P(B) = 0.2, and P(A and B) = 0.18, what is P(A or B)?
a) 0.38
b) 0.42
c) 0.56
10. If P(A) = 0.28, P(B) = 0.41, and P(A and B) = 0.16, what is P(A or B)?
a) 0.03
b) 0.44
c) 0.53
11. If P(A) = 0.27, P(B) = 0.47, and P(A and B) = 0.16, what is P(A or B)?
a) 0.38
b) 0.42
c) 0.56
12. If P(A) = 0.7, P(B) = 0.4, and P(A or B) = 0.8, what is P(A and B)?
a) 0.3
b) 0.4
c) 0.5
121
Day 3:
13. When two fair coins are tossed, there are four possible outcomes. What is the probability that two heads are tossed?
a)
1
8
b)
1
4
c)
3
8
d)
1
2
e)
2
3
14. When two fair coins are tossed, there are four possible outcomes. What is the probability that at least one tail is tossed?
a)
1
8
b)
1
4
c)
3
8
d)
1
2
e)
3
4
15. Events A and B are independent, P(A) = 0.7, P(B) = 0.4, what is P(A and B)?
a) 0.28
b) 0.30
c) 0.55
d) 0.72
e) 1.1
d) 0.4
e) 0.7
16. Events A and B are independent, P(A) = 0.2, P(B) = 0.5, what is P(A and B)?
a) 0.1
b) 0.3
c) 0.35
17. Events A and B are dependent, P(A) = 40%, and P(B | A) = 30%. What is P(A and B)?
a) 10%
b) 12%
c) 24%
d) 53%
e) 73%
18. Events A and B are dependent, P(A) = 15%, and P(B | A) = 60%. What is P(A and B)?
a) 9%
b) 38%
c) 45%
d) 75%
e) 90%
19. Events A and B are independent, P(A) = 26%, and P(B) = 42%. What is P(A and B)?
a) 11%
b) 16%
c) 26%
d) 34%
e) 68%
Day 4:
20. What is the probability of drawing all red cards from a standard deck of 52 cards if you draw 5 cards from the deck?
a) 0.0253
b) 0.253
c) 0.50
d) 0.523
e) 0.875
21. Three marbles are picked at random from a bag containing 4 red marbles and 5 white marbles. What is the probability that all
three marbles are red?
a)
•5 C 2
9 C3
4 C1
b)
• 5 C1
9 C3
4 C2
c)
•5 C 0
9 C3
4 C3
d)
•5 C3
9 C3
4 C0
e) None of these
22. Three marbles are picked at random from a bag containing 4 red marbles and 5 white marbles. What is the probability that exactly
2 marbles are red?
a)
•5 C 2
C
9 3
4 C1
b)
• 5 C1
C
9 3
4 C2
c)
•5 C 0
C
9 3
4 C3
d)
•5 C3
C
9 3
4 C0
e) None of these
23. Five cards are drawn at random from a standard deck. What is the probability that no aces are chosen?
a)
• 48 C 5
52 C 5
4 C0
b)
• 48 C1
52 C 5
4 C4
c)
• 4 C1
52 C 5
4 C4
d)
• 39 C1
52 C 5
13 C 4
e) None of these
24. Five cards are drawn at random from a standard deck. What is the probability that exactly 4 diamonds are chosen?
a)
• 48 C 5
52 C 5
4 C0
b)
• 48 C1
52 C 5
4 C4
c)
• 4 C1
52 C 5
4 C4
d)
• 39 C1
52 C 5
13 C 4
e) None of these
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25. Five prizes are being given in a raffle contest. 119 tickets are sold. After each prize is called, the winning ticket is returned to the
drawing box and is eligible to be picked for another prize. What is the probability that at least one of the tickets is drawn twice?
a) 0.0816
b) 0.0927
c) 0.1057
d) 0.8921
e) 0.9923
Day 5:
26. What is the probability of tossing a coin 10 times and getting exactly 4 heads?
a) 0.18
b) 0.21
c) 0.27
d) 0.32
e) 0.36
27. What is the probability of tossing a coin 20 times and getting exactly 16 tails?
a) 0.001
b) 0.005
c) 0.015
d) 0.160
e) 0.176
28. What is the probability that in a family of six children exactly four are boys? Assume a boy and a girl are equally likely.
a)
5
32
b)
11
64
c)
15
64
d)
1
3
e)
2
3
29. What is the probability that in a family of seven children exactly two are girls? Assume a boy and a girl are equally likely.
a)
7
128
b)
7
64
c)
21
128
d)
2
7
e)
5
7
30. What is the probability of 5 or more successes for a binomial experiment consisting of 7 trials with probability 0.25 of success on
each trial?
a) 0.0013
b) 0.0115
c) 0.0128
d) 0.0129
e) 0.0577
31. A baseball player’s batting average is 0.280. What is the probability that the player will get 3 or fewer hits in a game in which the
player has 5 official at-bats?
a) 0.114
b) 0.296
c) 0.376
d) 0.783
e) 0.976
Day 6:
32. The Nobel Prize for Physics is often given to more than one person. The probability distribution of the number of winners of the
Nobel Prize for Physics from 1901 – 2004 is given below. What is the expected number of winners?
Number of Winners n
Probability of n Winners
a) 0.03
b) 1.00
c) 1.53
1
0.480
2
0.265
3
0.255
d) 1.78
e) 1.98
33. The probability distribution of the number of piglets per litter on a farm is given below. What is the expected number of piglets per
litter on this farm?
Piglet Births per Litter
4
5
6
n piglets
0.2
0.4
0.4
Probability of n piglets
a) 4.8
b) 5.0
c) 5.2
d) 5.6
e) 5.8
Continued on next page….
123
For numbers 34 – 37, use the tree diagram to find each probability.
0.7
K′
K
0.3
J
34. P(K )
a) 0.21
0.9
J′
J
J′
b) 0.27
c) 0.3
d) 0.7
e) 0.9
b) 0.27
c) 0.49
d) 0.7
e) 0.8
b) 0.27
c) 0.49
d) 0.7
e) 0.8
b) 0.3
c) 0.49
d) 0.52
e) 1
d) 30
e) 32
35. P(J ′ | K)
a) 0.21
36. P(K and J ′ )
a) 0.21
37. P(J ′ )
a) 0.03
Previous Units Review:
38. What is the fourth term of the sequence defined by a n = 6n + 2 ?
a) 8
b) 20
c) 26
39. What is a rule for the nth term of the arithmetic sequence with a4 = 4 and a9 = 14?
a) a n = 2n − 4
b) a n = 4n − 6
c) a n = 4n + 6
d) a n = 2n + 4
e) a n = 4n − 2
40. What is a rule for the nth term of the geometric sequence 4, –8, 16, –32, 64, …?
a) a n = 2(− 4 )n −1
b) a n = −4(2 )n −1
c) a n = 4(− 2 )n −1
d) a n = −4(− 2 )n −1
e) a n = −2(4 )n −1
41. What is the sum of the first 6 terms of the geometric series 1 + 2 + 4 + 8 + 16 + …?
a) 63
b) 880
c) 1023
d) 1850
e) 4095
42. What is the common ratio of an infinite geometric series whose sum is 35 and the first term is a1 = 7?
4
d) 2
e) 8
5
43. In an activity club with 30 students, the offices of president, vice president, secretary, and treasurer will be filled. In how many
ways can the offices be filled?
a) –8
b) –2
c)
a) 4,060
b) 5,040
c) 27,405
d) 24,360
e) 657,720
44. How many different license plates are possible if three digits are followed by two letters?
a) 676,000
b) 876,450
c) 1,404,000
d) 1,757,600
e) 12,647,200
45. In how many ways can a 5 person committee be chosen from a group of 10 people?
a) 120
b) 252
c) 12,600
d) 30,240
e) 604,800
d) 90
e) 240
46. What is the coefficient of x4 in the expansion of (2x – 3)5?
a) –240
b) –90
c) 1
124
Pre-Calculus
Probability Unit Review Sheet
Name: _____________________________________
For numbers 1 – 4, suppose a card is drawn from a well-shuffled standard deck of 52 cards. Find the probability of drawing each of
the following.
1. a spade.
2. a black face card.
3. not a black jack.
4. a black diamond.
5. One of the integers between 11 and 20, inclusive, is picked at random. What is the probability that the integer is:
a) even?
b) divisible by 3?
c) a prime?
d) odd?
Use the following information for numbers 6 – 8: you have an equally likely chance of spinning any value on the spinner. Find the
probability of spinning the given event.
6. a factor of 27.
7. a prime number.
9
4
5
2
8. a two digit number.
9. A pet store has 46 dogs. Twenty-two are brown and 36 are white. Seventeen are both brown and white. Find the probability that a
randomly chosen dog will be white or brown.
10. A card is randomly selected from a standard deck of 52 cards. What is the probability that it is a three or a four?
11. A card is randomly selected from a standard deck of 52 cards. What is the probability that it is a face card or red?
For numbers 12 – 14, find the indicated probability.
12. P(A) = 0.25, P(B) = 0.2, P(A and B) = 0.15, P(A or B) = ________
125
13. P(A) =
2
1
1
, P(B) = , P(A and B) = ________, P(A or B) =
5
10
2
14. P(A) = 99%, P(A') = _______
For numbers 15 – 17, find the probability of randomly drawing the given marbles from a bag of 4 red, 6 green, and 2 blue marbles
(a) with replacement and (b) without replacement.
15. a red, then a green
16. a blue, then a red
17. a red, then a red
Use the following information for numbers 18 – 25: You have an equally likely chance of rolling any value on each of two dice. Find
the probability of the given event.
18. Rolling a sum of either 4 or 10.
19. Rolling a sum less than 9.
20. Rolling a sum greater than or equal to 9.
21. Rolling a sum that is not 8 or 10.
22. Rolling a sum that is less than 6 or odd.
23. Rolling a sum that is a multiple of 3 and a multiple of 4.
24. Rolling a sum that is a multiple of 5.
25. Rolling a sum that is not 3.
Use the following information for numbers 26 – 29: A jar contains 30 green marbles, 20 black marbles, and 10 yellow marbles. Find
the probability of choosing the given marbles from the jar (a) with replacement and (b) without replacement.
26. Green, then black.
27. Yellow, then yellow.
28. Green, then black, then yellow.
29. Black, then black, then green.
126
Use the following information for numbers 30 – 33: Four marbles are chosen at random from a jar containing 15 red marbles and 10
blue marbles.
30. What is the probability that the selection will contain at least one red marble?
31. What is the probability that the selection will contain at least one blue marble?
32. What is the probability that the selection will contain exactly one red marble?
33. What is the probability that the selection will contain exactly two blue marbles?
Use the following information for numbers 34 – 37: Three cards are drawn from a standard 52 card deck. Find the probability of the
following.
34. There are 2 kings and 1 other card.
35. All the cards are black.
36. There is 1 ace and 2 other cards.
37. Two of the cards are even and one is odd.
38. You randomly select two cards from a standard deck of 52 cards. What is the probability that the first card is a seven or an ace and
the second card is a seven, ace, or queen if you replace the first card before selecting the second?
39. You randomly select two cards from a standard deck of 52 cards. What is the probability that the first card is a seven or an ace and
the second card is a seven, ace, or queen if you do not replace the first card before selecting the second?
40. The letters of the word ABRACADABRA are written on separate slips of paper and placed in a hat. Five slips of paper are then
randomly drawn, one after the other and without replacement. What is the probability of getting:
a) all A’s?
b) both R’s?
c) at least one B?
d) one of each letter?
127
41. a) The letters of the word TUESDAY are put into a bag. If three letters are taken out and not replaced, what is the probability the
three letters are S, D, and E, in that order?
b) The following letters STODUN are put into a bag. If all the letters are taken out and not replaced, what is the probability that the
order you take them out spells DONUTS?
For numbers 42 – 47, use the tree diagram to the right to find each probability.
42. P(R ′ )
0.7
43. P(L | R)
R′
R
0.3
44. P(J and R )
0.2
45. P(R | J)
J
46. P(J and R ′)
0.5
K
L
J
0.4
K
L
47. P(R ′ | J)
For numbers 48 and 49, you toss a coin 3 times. Find the probability of the given event.
48. You toss exactly 1 tail.
49. You toss at least 1 tail.
For numbers 50 – 54, calculate the probability of tossing a coin 10 times and getting the given number of tails.
50. 3
51. 5
53. 6
54. 1
52. 9
55. Calculate the probability of randomly guessing at least 7 correct answers on a 10-question true-or-false quiz to get a passing grade.
56. Find the expected value:
x
P(x)
–4
0.2
–2
0.3
0
0.1
57. Find the expected value:
1
0.2
3
0.2
x
P(x)
–2
0.25
–1
0.35
0
0.05
1
0.15
3
0.2
128
Conic
Sections
129
130
Pre-Calculus
Conics Day 1 Notes: Parabolas
Vertex: the intersection of a parabola and its axis of symmetry.
Focus: the fixed point from which the locus of points on a parabola are
equidistant.
Parabola: the locus of points in a plane that are equidistant from a fixed point
and a specific line.
Conic section: a figure formed when a plane intersects a double-napped right cone.
Directrix: the line from which points on a parabola are equidistant.
Axis of symmetry: a line perpendicular to the directrix through the focus of a parabola.
You can use the standard form of the equation for a parabola to determine the characteristics of the parabola such as the vertex, focus,
and directrix.
Example 1: For the graph (y – 3)2 = –8 (x + 1), identify the vertex, focus, axis of symmetry, and directrix. Then graph the parabola.
131
Example 2: The parabolic mirror for the California Institute of Technology’s Hale telescope at Mount Palomar has a shape modeled
by y2 = 2668x, where x and y are measured in inches. What is the focal length of the mirror?
To determine the characteristics of a parabola, you may sometimes need to write an equation in standard form. In some cases, you can
simply rearrange the equation, but other times it may be necessary to use mathematical skills such as completing the square.
Example 3: Write x2 − 8x − y = −18 in standard form. Identify the vertex, focus, axis of symmetry, and directrix. Then graph the
parabola.
1
− x 2 + 3 x + 6 in standard form. Identify the vertex, focus, axis of symmetry, and directrix. Then graph the
Example 4: Write y =
2
parabola.
132
Example 5: Write an equation for and graph a parabola with the given characteristics.
a) focus (2, 1) and vertex (−5, 1)
b) vertex (3, −2), directrix y = −1
c) focus (−1, 7), opens up, contains (3, 7)
133
Line Tangent to a Parabola
A line ℓ that is tangent to a parabola at a point P forms an isosceles triangle such that:
*The segment from P to the focus forms one leg of the triangle.
*The segment along the axis of symmetry from the focus to another point on the tangent line forms
the other leg.
Example 6: Write an equation for the line tangent to y = x2 − 2 at (2, 2).
y2
5 − ; (1, − 4 ) .
Example 7: Write an equation for the line tangent to x =
5
134
Pre-Calculus
Conics Day 1: Parabolas Homework
Name: _____________________________________
For numbers 1 & 2, identify the vertex, focus, axis of symmetry, and directrix for each equation. Then graph the parabola.
1. (x – 1)2 = 8(y – 2)
2. y2 + 6y + 9 = 12 – 12x
Vertex: ______________
Vertex: ______________
Focus: _______________
Focus: _______________
Axis of symmetry: _______________
Axis of symmetry: _______________
Directrix: ___________________
Directrix: ___________________
3. Write x2 + 8x = –4y – 8 in standard form. Identify the vertex, focus, axis of symmetry, and directrix.
135
For numbers 4 & 5, write an equation for and graph a parabola with the given characteristics.
4. vertex (–2, 4); focus (–2, 3)
5. focus (2, 1); opens right; contains (8, –7)
Equation: ________________________________
Equation: ________________________________
6. SATELLITE DISH Suppose the receiver in a parabolic dish antenna is 2 feet from the vertex and is located at the focus. Assume
that the vertex is at the origin and that the dish is pointed upward. Find an equation that models a cross section of the dish.
7. REFLECTOR The figure shows a parabolic reflecting mirror. A cross section of the mirror can be modeled by 𝑥 2 = 16y, where the
values of x and y are measured in inches. Find the distance from the vertex to the focus of this mirror.
136
8. T-SHIRTS The cheerleaders at the high school basketball game launch T-shirts into the stands after a victory. The launching
device propels the shirts into the air at an initial velocity of 32 feet per second. A shirt’s distance y in feet above the ground after x
seconds can be modeled by y = –16x2 + 32x + 5.
a) Write the equation in standard form.
b) What is the maximum height that a T-shirt reaches?
9. FLASHLIGHT A flashlight contains a parabolic mirror with a bulb in the center as a light source and focus. If the width of the
mirror is 4 inches at the top and the height to the focus is 0.5 inch, find an equation of the parabolic cross section.
10. ARCHWAYS The entrance to a college campus has a parabolic arch above two columns
as shown in the figure.
a) Write an equation that models the parabola.
b) Graph the equation.
137
11. The cable for a suspension bridge is in the shape of a parabola. The vertical supports
are shown in the figure.
a) Write an equation for the parabolic cable.
b) Find the length of a supporting wire that is 100 feet from the center.
12. Write an equation for the line tangent to (x + 7)2 = −
13. Write an equation for the line tangent to=
y2
1
( y − 3) at (–5, –5).
2
1
( x − 4 ) at (24, 2).
5
14. Write an equation for the line tangent to –4x = (y + 5)2 at (0, –5).
For numbers 15 – 18, determine the orientation of each parabola.
15. directrix y = 4, p = –2
16. y2 = –8(x – 6)
17. vertex (–5, 1) , focus (–5, 3)
18. focus (7, 10), directrix x = 1
138
For numbers 19 – 22, write the equation for each parabola.
19.
20.
21.
22.
23. Abigail and Jaden are graphing x2 + 6x – 4y + 9 = 0. Is either of them correct? Explain your reasoning.
139
140
Pre-Calculus
Conics Day 2 Notes: Ellipses and Circles
An ellipse is the locus of points in a plane such that the sum of the
distances from two fixed points, called foci, is constant.
Major Axis: the segment that contains the foci of the ellipse and
has its endpoints on the ellipse.
Minor Axis: the segment through the center of an ellipse that is
perpendicular to the major axis and has endpoints on the ellipse.
Center: the midpoint of major and minor axes of an ellipse.
Vertices: the endpoints of the major axis of an ellipse.
Co-vertices: the endpoints of the minor axis of an ellipse.
Example 1: Graph the ellipse given by each equation.
a)
( x + 2)
9
2
+
( y − 1)
4
2
=
1
141
b) 4x2 + 24x + y2 – 10y – 3 = 0
Example 2: Write an equation for an ellipse with a major axis from (5, –2) to (–1, –2) and a minor axis from (2, 0) to (2, –4).
Example 3: Write an equation for an ellipse with vertices at (3, –4) and (3, 6) and foci at (3, 4) and (3, –2).
142
The eccentricity of an ellipse is the ratio of c to a. This value will always be between 0 and 1 and will determine how “circular” or
“stretched” the ellipse will be.
The value of c represents the distance between one of the foci and the center of the ellipse. As the foci are moved closer together, c
and e both approach 0. When the eccentricity reaches 0, the ellipse is a circle and both a and b are equal to the radius of the circle.
Example 4: Determine the eccentricity of the ellipse given by
( x − 4)
64
2
+
( y − 3)
36
2
=
1.
Example 5: The eccentricity of the orbit of Uranus is 0.47. Its orbit around the Sun has a major axis length of 38.36 AU (astronomical
units). What is the length of the minor axis of the orbit?
Example 6: A lake in a park is elliptically-shaped. If the length of the lake is 2500 meters and the width is 1500 meters, find the
eccentricity of the lake.
143
144
Pre-Calculus
Conics Day 2: Ellipses and Circles Homework
Name: _____________________________________
For numbers 1 & 2, graph the ellipse given by each equation.
1. 4x2 + 9y2 – 8x – 36y + 4 = 0
2. 25x2 + 9y2 – 50x – 90y + 25 = 0
Equation: ___________________________
Equation: ___________________________
Foci: _______________________________
Foci: _______________________________
Vertices: ____________________________
Vertices: ____________________________
Co-Vertices: _________________________
Co-Vertices: _________________________
For numbers 3 & 4, write an equation for the ellipse with each set of characteristics.
3. vertices (–12, 6), (4, 6); foci (–10, 6), (2, 6)
4. foci (–2, 1), (–2, 7); length of major axis 10 units
145
For numbers 5 – 8, write each equation in standard form. Identify the related conic.
5. y2 – 4y = 4x + 16
6. 4x2 – 32x + 3y2 – 18y = –55
7. x2 + y2 – 8x – 24y = 9
8. x2 + y2 + 20x – 10y + 4 = 0
For numbers 9 & 10, determine the eccentricity of the ellipse given by each equation.
9.
( x + 1)
25
2
+
( y + 1)
16
2
=
1
10.
( y + 2)
64
2
+
( x + 1)
9
2
1
=
11. CONSTRUCTION A semi-elliptical arch is used to design a headboard for a bed frame. The headboard will have a height of 2
feet at the center and a width of 5 feet at the base. Where should the craftsman place the foci in order to sketch the arch (In other
words, find the location of the foci)?
12. WHISPERING GALLERY A whispering gallery at a museum is in the shape of an ellipse. The room is 84 feet long and 46 feet
wide.
a) Write an equation modeling the shape of the room. Assume that it is centered at the origin and that the major axis is horizontal.
b) Find the location of the foci.
146
13. SIGNS A sign is in the shape of an ellipse. The eccentricity is 0.60 and the length is 48 inches.
a) Write an equation for the ellipse if the center of the sign is at the origin and the major axis is horizontal.
b) What is the maximum height of the sign?
14. TUNNEL The entrance to a tunnel is in the shape of half an ellipse as shown in the figure.
(You may assume that the ellipse is centered at the origin)
a) Write an equation that models the ellipse.
b) Find the height of the tunnel 10 feet from the center.
15. RETENTION POND A circular retention pond is getting larger by overflowing
and flooding the nearby land at a rate that increases the radius 100 yards per day,
as shown below.
a) Graph the circle that represents the water, and find the distance from the center
of the pond to the house.
b) If the pond continues to overflow at the same rate, how many days will it take for the water to reach the house?
147
c) Write an equation for the circle of water at the current time and an equation for the circle when the water reaches the house.
For numbers 16 – 19, write an equation for each ellipse.
16.
17.
18.
19.
For numbers 20 – 22, write the standard form of the equation for each ellipse.
3
20. The vertices are (–10, 0) and (10, 0), and the eccentricity e is .
5
4
21. The co-vertices are (0, 1) and (6, 1), and the eccentricity e is .
5
(
)
22. The center is at (2, –4), one focus is at 2, − 4 + 2 5 , and the eccentricity e is
5
.
3
148
Pre-Calculus
Conics Day 3 Notes: Hyperbolas
Hyperbola: the set of all points in a plane such that the absolute value
of the differences of the distances from two foci is constant.
Transverse axis: a segment that has a length of 2a units and connects
the vertices.
Conjugate Axis: the segment that is perpend icular to the transverse axis,
passes through the center, and has length of 2b units.
The general form for a hyperbola centered at (h, k) is given below.
149
Example 1: Graph the hyperbola given by
Example 2: Graph the hyperbola given by
x2 y 2
−
=
1.
49 81
( y + 4)
4
2
−
( x − 2)
9
2
=
1.
150
Example 3: Graph the hyperbola given by 4x2 – y2 + 24x + 4y = 28.
You can determine the equation for a hyperbola if you are given characteristics that provide sufficient information.
Example 4: Write an equation for the hyperbola with foci (1, –5) and (1, 1) and transverse axis length of 4 units.
Example 5: Write an equation for the hyperbola with vertices (–3, 10) and (–3, –2) and conjugate axis length of 6 units.
151
Another characteristic that can be used to describe a hyperbola is the eccentricity. The formula for eccentricity is the same for all
c
conics, e = . Recall, that for an ellipse, the eccentricity is greater than 0 and less than 1. For a hyperbola, the eccentricity will
a
always be greater than 1.
Example 6: Find the eccentricity of
( y + 2)
32
2
−
( x − 1)
25
2
=
1.
You can determine the type of conic when the equation for the conic is in general form, Ax2 + Bxy + Cy2 + Dx + Ey + F = 0. The
discriminant, or B2– 4AC, can be used to identify the conic.
Example 7: Use the discriminant to identify the conic section in the equation 2x2 + y2 – 2x + 5xy + 12 = 0.
Example 8: Use the discriminant to identify the conic section in the equation 4x2 + 4y2 – 4x + 8 = 0.
Example 9: Use the discriminant to identify the conic section in the equation 2x2 + 2y2 – 6y + 4xy – 10 = 0.
152
Example 10: LORAN (LOng RAnge Navigation) is a navigation system for ships relying on radio pulses that is not dependent on
visibility conditions. Suppose LORAN stations E and F are located 350 miles apart along a straight shore with E due west of F. When
a ship approaches the shore, it receives radio pulses from the stations and is able to determine that it is 80 miles farther from station F
than it is from station E. Find the equation for the hyperbola on which the ship is located.
Example 11: LORAN (LOng RAnge Navigation) is a navigation system for ships relying on radio pulses that is not dependent on
visibility conditions. Suppose LORAN stations E and F are located 350 miles apart along a straight shore with E due west of F. When
a ship approaches the shore, it receives radio pulses from the stations and is able to determine that it is 80 miles farther from station F
than it is from station E. Find the exact coordinates of the ship if it is 125 miles from the shore.
153
154
Pre-Calculus
Conics Day 3: Hyperbolas Homework
Name: _____________________________________
For numbers 1 & 2, graph the hyperbola given by each equation.
y 2 ( x − 1)
−
=
1
16
4
2
1. x2 – 4y2 – 4x + 24y – 36 = 0
2.
For numbers 3 – 8, write an equation for the hyperbola with the given characteristics.
3. vertices (–10, 6), (4, 6); foci (–12, 6), (6, 6)
4. foci (0, 6), (0, –4); length of transverse axis 8 units
5. foci (–1, 9), (–1, –7); conjugate axis length of 14 units
3
45
6. vertices (–1, 9), (–1, 3) asymptotes y =
± x+
7
7
7
59
7. center (–7, 2); asymptotes y =
± x + , transverse axis
5
5
length of 10 units
19
8. center (0, –5); asymptotes y =
±
x − 5, conjugate axis
6
length of 12 units
155
9. Determine the eccentricity of the hyperbola given by the equation
( x − 7)
36
2
−
( y + 10 )
121
2
=
1.
10. ENVIRONMENTAL NOISE Two neighbors who live one mile apart hear an explosion while they are talking on the telephone.
One neighbor hears the explosion two seconds before the other. If sound travels at 1100 feet per second, determine the equation of the
hyperbola on which the explosion was located.
For numbers 11 – 14, use the discriminant to identify each conic section.
11. 5x2 + xy + 2y2 – 5x + 8y + 9 = 0
12. 16x2 – 4y2 – 8x – 8y + 1 = 0
13. 4x2 + 8xy + 4y2 + x + 11y + 10 = 0
14. 2x2 + 4y2 – 3x – 6y + 2 = 0
15. EARTHQUAKES The epicenter of an earthquake lies on a branch of the hyperbola represented by
( x − 50 )
1600
2
−
( y − 35)
2500
2
=
1,
where the seismographs are located at the foci.
a) Graph the hyperbola.
156
b) Find the locations of the seismographs.
16. SHADOWS A lamp projects light onto a wall in the shape of a hyperbola. The edge of the light can be modeled by
y2
x2
−
=
1.
196 121
a) Graph the hyperbola.
b) Write the equations of the asymptotes.
c) Find the eccentricity.
17. PARKS A grassy play area is in the shape of a hyperbola, as shown.
a) Write an equation that models the curved sides of the play area.
b) If each unit on the coordinate plane represents 3 feet, what is the narrowest vertical width of the play area?
157
18. SHADOWS The path of the shadow cast by the tip of a sundial is usually a hyperbola.
a) Write two equations of the hyperbola in standard form if the center is at the origin, given that the path contains a transverse axis of
24 millimeters with one focus 14 millimeters from the center.
b) Graph one hyperbola.
For numbers 19 – 22, solve each system of equations. Round to the nearest tenth if necessary.
2 y x and
19.=
21.
( x − 3)
16
2
−
( y + 2)
2
= 1
84
y 2 x2
y 2 x2
+ =1 and
−
=
1
36 25
36 25
1
x2 ( y − 4)
20. y =
− x + 3 and
−
=
1
4
36
4
2
( x + 1) ( y + 2 )
x2 y 2
22. =
−
1 and
−= 1
4
1
49
4
2
2
158
23. ASTRONOMY While each of the planets in our solar system move around the Sun in elliptical orbits, comets may have elliptical,
parabolic, or hyperbolic orbits where the center of the sun is a focus.
The paths of three comets are modeled below, where the values of x and y are measured in gigameters. Use the discriminant to
identify each conic.
a) 3x2 – 18x – 580850 = 4.84y2 – 38.72y
b) –360x – 8y = –y2 – 1096
c) –24.88y + x2 = 6x – 3.11y2 + 412341
For numbers 24 & 25, write an equation for each hyperbola.
24.
25.
159
160
Pre-Calculus
Conics Day 4 Notes: Parametric Equations
Consider the graphs below, each of which models different aspects of what happens when a certain object is thrown into the air.
Figure (a) shows the vertical distance the object travels as a function of time, while Figure (b) shows the object’s horizontal distance
as a function of time. Figure (c) shows the object’s vertical distance as a function of its horizontal distance.
(a)
(b
(c)
Each of these graphs and their equations tells part of what is happening in this situation, but not the whole story. To express the
position of the object, both horizontally and vertically, as a function of time we can use parametric equations. The equations below
both represent the graph shown in Figure (c).
Rectangular Equation
2 2
y =−
x + x + 40
225
Parametric Equations
x = 30 2t
Horizontal component
−16t 2 + 30 2t + 40
y=
Vertical component
From the parametric equations, we can now determine where the object
was at a given time by evaluating the horizontal and vertical components
for t. For example, when t = 0, the object was at (0, 40). The variable t is
called a parameter.
The graph shown plotted over the time interval 0 ≤ t ≤ 4. Plotting points
in the order of increasing values of t traces the curve in a specific direction
called the orientation of the curve. This orientation is indicated by arrows
on the curve as shown.
161
Example 1: Sketch the curve created by x = t2 – 1 and y=
Example 2: Sketch the curve created by =
x
t
+ 2 over the interval –3 ≤ t ≤ 3.
4
t
t2
+ 2 over the interval –5 ≤ t ≤ 5.
− 1 and y=
5
4
Example 3: Write y = 2t and x = t2 + 2 in rectangular form.
162
Example 4: Write =
y
1
and =
x
2t
t + 1 in rectangular form. Then graph the equation. State any restrictions on the domain.
The parameter in a parametric equation can also be an angle θ.
Example 5: Write y = 5 sin θ and x = 3 cos θ in rectangular form. Then graph the equation.
163
Example 6: Use the parameter t = x – 1 to write the parametric equations that can represent y = x2 + 2. Then graph the equation,
indicating the speed and orientation.
Example 7: Use the parameter t = 2x to write the parametric equations that can represent y = x2 + 2. Then graph the equation,
indicating the speed and orientation.
Example 8: Use the parameter t= 2 −
x
to write the parametric equations that can represent y = x2 + 2. Then graph the equation,
2
indicating the speed and orientation.
164
Parametric equations are often used to simulate projectile motion. The path of a projectile launched at an angle other than 90° with the
horizontal can be modeled by the following parametric equations.
Example 9: Shane Lechler of the Oakland Raiders has the record career punting average with an average of 46.47 yards. Suppose that
he kicked the ball with an initial velocity of 16 yards per second at an angle of 72°. How far will the ball travel horizontally, after 2
seconds, if he punts it with an initial height of 2 feet?
Example 10: Aaron kicked a soccer ball with an initial velocity of 39 feet per second at an angle of 44˚ with the horizontal. After 0.9
second, how far has the ball traveled horizontally?
165
166
Pre-Calculus
Conics Day 4: Parametric Equations Homework
Name: _____________________________________
For numbers 1 & 2, sketch the curve given by each pair of parametric equations over the given interval.
1. x = t2 + 1 and y=
t
− 6; –5 ≤ t ≤ 5
2
2. x = 2t + 6 and y = −
t2
; –5 ≤ t ≤ 5
2
For numbers 3 – 6, write each pair of parametric equations in rectangular form.
3. x = 2t + 3, y = t – 4
4. x = t + 5, y = –3t2
5. x = 3 sin θ, y = 2 cos θ
6. y = 4 sin θ, x = 5 cos θ
167
For numbers 7 & 8, use each parameter to write the parametric equations that can represent each equation. Then graph the equation,
indicating the speed and orientation.
=
7. t
2− x
3 − x2
=
for y
3
2
8. t = 4x – 1 for y = x2 + 2
9. MODEL ROCKETRY Manuel launches a toy rocket from ground level with an initial velocity of 80 feet per second at an angle of
80° with the horizontal.
a) Write parametric equations to represent the path of the rocket.
b) How long will it take the rocket to travel 10 feet horizontally from is starting point? What will be its vertical distance at that point?
10. PHYSICS A rock is thrown at an initial velocity of 5 meters per second at an angle of 8° with the ground. After 0.4 second, how
far has the rock traveled horizontally?
11. PLAYING CATCH Tom and Sarah are playing catch. Tom tosses a ball to Sarah at an initial velocity of 38 feet per second at an
angle of 28° from a height of 4 feet. Sarah is 40 feet away from Tom.
a) How high above the ground will the ball be when it gets to Sarah?
b) What is the maximum height of the ball?
168
12. TENNIS Melinda hits a tennis ball with an initial velocity of 42 feet per second at an angle of 16° with the horizontal from a
height of 2 feet. She is 20 feet from the net and the net is 3 feet high. Will the ball go over the net?
15. BASKETBALL Mandy throws a basketball with an initial velocity of 28 feet per second at an angle of 60° with the horizontal. If
Mandy releases the ball from a height of 5 feet, write a pair of equations to determine the vertical and horizontal positions of the ball.
16. GOLF Julio hit a golf ball with an initial velocity of 100 feet per second at an angle of 39° with the horizontal.
a) Write parametric equations for the flight of the ball.
b) Find the maximum height the ball reaches.
17. BASEBALL Micah hit a baseball at an initial velocity of 120 feet per second from
a height of 3 feet at an angle of 34°.
a) How far will the ball travel horizontally before it hits the ground?
b) What is the maximum height the ball will reach?
c) If the fence is 8 feet tall and 400 feet from home plate, will the ball clear the fence to be a home run? Explain.
169
170
Trig Unit 1
171
172
Trig Day 1 Notes
Recall that an angle is formed by two rays that have a common endpoint, called the vertex. You can generate any angle by fixing one
ray, called the initial side, and rotating the other ray, called the terminal side, about the vertex. In a coordinate plane, an angle whose
vertex is at the origin and whose initial side is the positive x-axis is in standard position.
The measure of an angle is determined by the amount and direction of rotation from the initial side to the terminal side. The angle
measure is positive if the rotation is counterclockwise, and negative if the rotation is clockwise. The terminal side of an angle can
make more than one complete rotation.
Quadrant I
*The common unit to measure very large angles is the revolution.
*The common unit to measure smaller angles is the degree.
Quadrant II
Quadrant III
Quadrant IV
Example 1: Draw an angle with the given measure in standard position. Then tell which quadrant the terminal side lies.
a) –120°
b) 400°
c) –525°
d) 240°
*Two angles in standard position are called coterminal angles if they have the same terminal ray.
To find a positive coterminal angle add 360° (or 2π) to the original angle until the answer is positive.
To find a negative coterminal angle subtract 360° (or 2π) from the original angle until the answer is negative.
Example 2: Find one positive and one negative angle that are coterminal with
a) –100°
b) 575°
So far, all the angles you have worked with have been measured in degrees. You can also measure angles in radians. To define a
radian, consider a circle with radius r centered at the origin. One radian is the measure of an angle in standard position whose terminal
side intercepts an arc of length r.
173
Because the circumference of a circle is 2πr, there are 2π radians in a full circle. Degree measure and radian measure are therefore
related by the equation 360° = 2π radians, or 180° = π radians.
The diagram shows equivalent radian and degree measures
for special angles from 0° to 360° (0 radians to 2π radians).
You may find it helpful to memorize the equivalent degree and radian measures of special angles in the first quadrant and for 90° =
π
2
radians. All other special angles are just multiples of these angles.
You can also use the following rules to convert degrees to radians and radians to degrees.
Example 3:
b) Convert −
a) Convert 320° to radians. Round to the nearest hundredth.
5π
radians to degrees. Round to the
12
nearest tenth.
c) Convert 3.7 radians to degrees. Round to the nearest hundredth.
Example 4: Find one positive and one negative angle that are coterminal with
a) −
2π
3
b)
17π
4
174
Pre-Calculus
Trig Day 1 Worksheet
Name: _____________________________________
For numbers 1 – 3, match the radian measure with its’ corresponding degree measure.
1.
5π
6
2.
a) 330°
3π
4
3.
b) 150°
11π
6
c) 135°
For numbers 4 – 7, convert each angle in degrees to radians. Leave your answer in terms of π.
4. 18°
5. 175°
6. 230°
7. –140°
13π
7
11. −4π
For numbers 8 – 11, convert each angle in radians to degrees.
8.
π
9
9.
3π
5
10.
For numbers 12 – 14, convert each angle in degrees to radians. Round to two decimal places.
12. 76°
13. –50°
14. 250°
For numbers 15 – 17, convert each angle in radians to degrees. Round to two decimal places.
15. 3 radians
16.
π
radians
17
17. –5.2 radians
For numbers 18 – 21, draw each angle in standard position and then state which quadrant the angle lies.
18.
3π
4
19. −
5π
4
20. 120°
21. 420°
For numbers 22 – 24, find one positive and one negative coterminal angle.
22. 415°
23.
19π
6
24. −
π
50
175
176
Pre-Calculus
Trig Day 1 Homework
Name: _____________________________________
For numbers 1 – 3, match the radian measure with its’ corresponding degree measure.
4π
3
1.
2.
a) 315°
3π
2
3.
b) 240°
7π
4
c) 270°
For numbers 4 – 7, convert each angle in degrees to radians. Leave your answer in terms of π.
4. 145°
5. 125°
6. 800°
7. –215 °
For numbers 8 – 11, convert each angle in radians to degrees.
8.
5π
8
9.
5π
9
10.
4π
5
11. −3π
For numbers 12 – 14, convert each angle in degrees to radians. Round to two decimal places.
12. 18°
13. –40°
14. 200°
For numbers 15 – 17, convert each angle in radians to degrees. Round to two decimal places.
15. 2 radians
16.
π
radians
13
17. –4.8 radians
For numbers 18 – 21, draw each angle in standard position and then state which quadrant the angle lies.
18.
7π
6
19. −
2π
3
20.
16π
3
21. –210°
177
For numbers 22 – 27, find one positive and one negative coterminal angle.
22. 395°
25.
π
4
23π
5
23. –765°
24.
26. 315°
27. −
23π
5
178
Trig Day 2 Notes
Sector: a region of a circle that is bounded by two radii and an arc of the circle.
Central angle: an angle formed by two radii of a circle.
Sector
Arc Length
Area
Radians
s = rθ
Degrees
A=
A=
s=
Example 1:
a) Find the arc length and area of a sector with a radius of 5 centimeters and a central angle of
π
4
. Round answers to the nearest tenth.
b) Find the arc length and area of a sector with radius 12 in. and a central angle of 120°. Leave answer in terms of π.
c) A carousel with a diameter of 25 feet takes 16 seconds to make one rotation. If you ride the carousel for 2 minutes, through what
angle do you rotate? How many feet does a point on the outer edge of the carousel revolve?
Example 2: A sector of a circle has area 88 cm2 and central angle 0.4 radians. Find its radius and arc length. Round answers to the
nearest tenth.
179
Example 3: The latitude of Durham, North Carolina, is 36°N. About how far from Durham is the North Pole? (The radius of the
earth is about 3963 miles) Round answer to the nearest tenth.
Example 4: Beijing, China, is due north of Perth, Australia. The latitude of Beijing is 39.92°N and the latitude of Perth is 31.97°S.
About how far apart are the two cities? (The radius of the earth is about 3963 miles) Round answer to the nearest hundredth.
180
Pre-Calculus
Trig Day 2 Worksheet
Name: _____________________________________
1. A sector of a circle has radius 9 cm and central angle 0.8 radians. Find its arc length and area. Round answers to the nearest tenth.
2. A sector of a circle has radius 10 cm and central angle 80˚. Find its arc length and area. Round answers to the nearest tenth.
3. A sector of a circle has arc length 22 cm and central angle 5.3 radians. Find its radius and area. Round answers to the nearest tenth.
4. A sector of a circle has arc length 6 cm and central angle 59˚. Find it radius and area. Round answers to the nearest tenth.
5. A sector of a circle has area 35 cm2 and central angle 0.7. Find its radius and arc length. Round answers to the nearest tenth.
6. A sector of a circle has radius 5 cm and central angle 137°. Find it approximate arc length and area. Round answers to the nearest
tenth.
7. A sector of a circle has central angle 70° and arc length 7.5 cm. Find its area to the nearest square centimeter.
8. A sector of a circle has central angle 48° and arc length 22.2 cm. Find its area to the nearest square centimeter.
9. The latitude of Lima, Peru, is 12°S. About how far from Lima is the South Pole? (The radius of the earth is about 3963 miles)
Round answer to the nearest hundredth.
181
182
Pre-Calculus
Trig Day 2 Homework
Name: _____________________________________
1. A sector of a circle has radius 6 cm and central angle of 50˚. Find its arc length and area. Round answers to the nearest tenth.
2. A sector of a circle has radius 5 cm and central angle 3 radians. Find its arc length and area. Round answers to the nearest tenth.
3. A sector of a circle has arc length 11 cm and central angle 2.2 radians. Find its radius and area. Round answers to the nearest tenth.
4. A sector of a circle has arc length 2 cm and central angle 34˚. Find its radius and area. Round answers to the nearest tenth.
5. A sector of a circle has area 25 cm2 and central angle 29˚. Find its radius and arc length. Round answers to the nearest tenth.
6. A sector of a circle has area 90 cm2 and central angle 0.2 radians. Find its radius and arc length. Round answers to the nearest tenth.
7. A sector of a circle has central angle 30° and arc length 3.5 cm. Find its area to the nearest square centimeter.
8. A sector of a circle has central angle
2π
and arc length 8.4 cm. Find its area to the nearest square centimeter.
15
183
9. Memphis, Tennessee, is due north of New Orleans, Louisiana. The latitude of Memphis is 35.1°N and the latitude of New Orleans
is 30°N. About how far apart are the two cities? (The radius of the earth is about 3963 miles) Round answer to the nearest hundredth.
10. The minute hand of a clock moves from 12 to 2 o’clock, or
1
of a complete revolution. Through how many degrees does it
6
move? Through how many radians does it move?
11. The minute hand of a clock is 8 inches long and moves from 12 to 2 o’clock. How far does the tip of the minute hand move?
Round the answer to two decimal places.
12. What is the arc length of a sector with a radius of 5 cm and a central angle of 20°?
a) 1.75 cm
b) 2.8 cm
c) 4.35 cm
d) 4 cm
e) 100 cm
13. What is the arc length of a sector with a radius of 7.3 in. and a central angle of 66°?
a) 6.7 in.
b) 7.3 in.
c) 8.4 in.
d) 8.9 in.
e) 10.5 in.
184
Trig Day 3 Notes
Consider a right triangle, one of whose acute angles is θ (the Greek letter theta). The three sides of the
triangle are the hypotenuse, the side opposite θ, and the side adjacent to θ.
hypotenuse
Ratios of a right triangle’s three sides are used to define the six trigonometric functions: sine (sin),
cosine (cos), tangent (tan), secant (sec), cosecant (csc), and cotangent (cot).
opposite side
θ
adjacent side
Right Triangle Definition of Trigonometric Functions
Let θ be an acute angle of a right triangle. The six trigonometric functions of θ are defined as follows:
sin θ
=
opp
adj
opp
cos θ =
tan θ
=
hyp
hyp
adj
csc θ
=
hyp
hyp
adj
sec θ =
cot θ
=
opp
adj
opp
SOH CAH TOA
The abbreviations opp, adj, and hyp represent the lengths of the three sides of the right triangle. Note that the ratios in the second row
are the reciprocals of the ratios in the first row. That is:
csc θ =
1
sin θ
sec θ =
1
cos θ
cot θ =
1
tan θ
Example 1:
a) Evaluate the six trigonometric functions of
b) Find the value of x for the right triangle shown.
the angle θ shown in the right triangle.
30°
θ
15
13
5
x
12
c) Solve ∆ABC.
B
62°
c
A
6
b
C
185
Example 2:
a) You are flying a kite 4 feet above the ground using 300 feet of line. With a wind speed of 40 miles per hour, the angle the kite line
makes with the ground is 29°. How high is the kite flying from the ground?
b) An airplane flying at 20,000 feet is headed toward an airport. The airport’s landing systems sends radar signals from the runway to
the airplane at a 5° angle of elevation. How many miles (measured along the ground) is the airplane from the runway?
(1 mile = 5,280 feet)
Example 3: Evaluate each expression. If necessary, express the value without a square root in the denominator by rationalizing.
a) cos 45°
b) tan
π
4
Example 4: Use a calculator to find the value of the trigonometric function to four decimal places.
a) cot 90°
b) sec
π
6
186
Pre-Calculus
Trig Day 3 Worksheet
Name: _____________________________________
For numbers 1 – 4, use the given right triangles to evaluate each expression. If necessary, express the value without the square root in
the denominator by rationalizing.
1. tan 30°
45°
2. csc 45°
30°
1
2
45°
3. cot
π
3
4. tan
π
4
+ csc
π
1
60°
6
1
5. Find the measure of the side of the right triangle whose length is designated by a lowercase letter. Round answers to the nearest
whole number.
B
221 in.
a
34°
A
C
b
For numbers 6 – 8, use a calculator to find the value of the trigonometric function to four decimal places.
6. tan 32.7°
7. cos
π
10
8. sec
11π
36
9. At a point on the ground 50 feet from the foot of a tree, the angle of elevation to the top of the tree is 53º. Find the height of the
tree.
187
10. From the top of a lighthouse 210 feet high, the angle of depression to a boat is 27º. Find the distance from the boat to the foot of
the lighthouse. The lighthouse was built at sea level.
11. From the top of a lighthouse 160 feet high, the angle of depression of a boat out at sea is an angle of 24°. Find to the nearest foot
the distance from the boat to the foot of the lighthouse, the foot of the lighthouse being at sea level.
188
Pre-Calculus
Trig Day 3 Homework
Name: ______________________________
For numbers 1 – 4, use the given right triangles to evaluate each expression. If necessary, express the value without the square root in
the denominator by rationalizing.
45°
1. cos 30°
30°
2. sec 45°
1
2
45°
3. tan
π
4. sin
3
π
4
– sin
1
π
60°
4
1
For numbers 5 and 6, find the measure of the side of the right triangle whose length is designated by a lowercase letter. Round
answers to the nearest whole number.
B
5.
6.
B
16 m
a
A
37°
250 cm
C
c
23°
A
C
For numbers 7 – 9, use a calculator to find the value of the trigonometric function to four decimal places.
7. sin 38°
8. csc 17°
9. cot
π
12
10. Richard is flying a kite. The kite string makes an angle of 57º with the ground. If Richard is standing 100 feet from the point on
the ground directly below the kite, find the length of the kite string.
189
11. From the top of a tower 80 feet high, the angle of depression of an object on the ground contains 38°. Find to the nearest foot, the
distance from the object to the top of the tower.
12. At a point on the ground 50 feet from the foot of a tree, the angle of elevation of the top of the tree contains 48°. Find the distance
from the point on the ground to the top of the tree to the nearest foot.
13. In order to construct a bridge, the width of the river must be determined. Suppose a stake is planted on one side of the river
directly across from a second stake on the opposite side. At a distance 50 meters to the left of the stake, an angle of 82 degrees is
measures between the two stakes. Find the width of the river.
14. The aerial run in Snowbird, Utah has an angle of elevation of 20.2 degrees. Its vertical drop is 2900 feet. Estimate the length of
this run.
15. Ace wants to build a rope bridge between his tree house and Willard’s tree house. Suppose Ace’s tree house is directly behind
Willard’s tree house. At a distance of 20 meters to the left of Ace’s tree house, an angle of 52 degrees is measured between the two
tree houses. Find the length of the rope bridge.
16. A ramp for unloading a moving truck has an angle of elevation of 32 degrees. If the top of the ramp is 4 feet above the ground,
estimate the length of the ramp.
17. John found two trees directly across from each other in a canyon. When he moved 100 feet from the tree on his side, the angle
formed by the tree on his side, John, and the tree on the other side was 70 degrees. Find the distance across the canyon.
190
Pre-Calculus
Quiz Review Days 1 – 3
Name: _____________________________________
1. The minute hand of a clock is 6 inches long and moves from 12 to 5 o’clock. How far does the tip of the minute hand move?
Round to two decimal places.
2. The minute hand of a clock is 2 feet long and moves from 12 to 7 o’clock. How far does the tip of the minute hand move? Round
to two decimal places.
3. Convert to radian measure (leave answer in π form): 78°
4. Convert to degrees: −
7π
16
Find two angles, one positive and one negative that are coterminal with each given angle.
5. –430°
6.
5π
4
Use the following information for questions 7 & 8: A sector of a circle has a radius of 2 mm and a central angle of 4 radians.
7. What is the arc length of the given sector?
8. What is the area of the given sector?
191
For numbers 9 and 10, find the measure of the side of the right triangle whose length is designated by a lowercase letter. Round
answers to the nearest whole number.
B
9.
10.
B
55°
a
32 m
C
A
114 m
c
16°
A
C
11. When landing, a jet will average a 3 degree angle of descent. What is the altitude to the nearest foot, of a jet on final descent as it
passes over an airport beacon 6 miles from the start of the runway?
12. In a sightseeing boat near the base of the Horseshoe Falls at Niagara Falls, a passenger estimates the angle of elevation to the top
of the falls to be 30 degrees. If the Horseshoe Falls are 173 feet high, what is the distance from the boat to the base of the falls?
192
Trig Day 4 Notes
If θ is a real number and P = (x, y) is a point on the unit circle that corresponds to θ, then
sin θ = y
(0, 1)
cos θ = x
(–1, 0)
y
=
θ
tan
,x ≠ 0
x
(1, 0)
(0, –1)
Example 1: A point P(x, y) is shown on the unit circle corresponding to a real number t. Find the values of the six trigonometric
functions at t.
If the circle is not a unit circle then the following is true:
sin θ =
y
r
csc θ =
r
,y≠0
y
cos θ =
x
r
sec θ =
r
,x≠0
x
tan θ =
y
,x≠0
x
cot θ =
x
,y≠0
y
Evaluating the 6 trig functions using a point on a terminal side…
Use the Pythagorean Theorem to find the 3rd side (r).
θ is always the vertex angle.
Example 2: Use the given point on the terminal side of an angle θ in standard position. Evaluate the six trig functions of θ.
a) (–8, 15)
b) (1, – 3 )
193
Example 3: Name the quadrant described in the following:
a) sin θ > 0 and cos θ > 0
b) sin θ < 0 and sec θ > 0
S
A
T
C
Example 4: Without using a calculator or table, solve each equation for all θ in radians.
a) cos θ = 1
b) sin θ = –1
c) cos θ = 0
Example 5: Without using a calculator or table, state whether each expression is positive, negative, or zero.
a) sin 5π
b) cos
Example 6: If tan θ < 0 and sin θ = −
Example 7: If csc θ = −
11π
6
c) cos 164°
8
, find the five remaining trigonometric functions.
17
17
π
3π
and ≤ θ ≤
, find the five remaining trigonometric functions.
15
2
2
Example 8: Find:
a) cos 180°
b) sin 720°
c) cos
3π
2
194
195
196
Pre-Calculus
Trig Day 4 Homework
Name: _____________________________________
1. If tan θ > 0 and sin θ = −
9
, find the five remaining trigonometric functions.
41
2. If sin θ > 0 and cos θ = −
11
, find the five remaining trigonometric functions.
61
3. If cot θ = −
4. If sec θ =
40
and 0 ≤ θ ≤ π , find the five remaining trigonometric functions.
9
13
and π ≤ θ ≤ 2π , find the five remaining trigonometric functions.
12
For numbers 5 & 6, a point P(x, y) is shown on the unit circle corresponding to a real number t. Find the values of the six
trigonometric functions at t.
y
5.
y
6.
P
t
t
O
(1, 0)
x
O
(1, 0)
x
P
197
For numbers 7 – 9, use the given point on the terminal side of an angle θ in standard position to evaluate the six trigonometric
functions.
7. (–12, 5)
8. (15, –8)
(
9. − 15, 5 7
)
For numbers 10 – 12, name the quadrant described in the following:
10. tan θ > 0 and cos θ < 0
11. sin θ < 0 and sec θ > 0
12. cot θ < 0 and csc θ > 0
For numbers 13 – 15, without using a calculator or table, solve each equation for all θ in radians.
13. cos θ = –1
14. sin θ = 0
15. tan θ = 0
For numbers 16 – 18, without using a calculator or table, state whether each expression is positive, negative, or zero.
16. sin 8π
17. cos
4π
3
18. cos 164°
For numbers 19 – 21, find the following:
19. cos –90°
20. tan 540°
21. sin 7π
198
Trig Day 5 Notes
Reference Angles
Let θ be an angle in standard position. Its’ reference angle is the acute positive angle θ ´ (theta prime) formed by the terminal side of
θ and the x-axis.
Reference Angle Chart:
This chart shows the relationship between θ and θ ´ for nonquadrantal angles, such that 90° < θ < 360°
90° < θ < 180°; θ′ = 180° – θ
180° < θ < 270°; θ′ = θ – 180°
270° < θ < 360°; θ′ = 360° – θ
θ′
θ′
θ′
Example 1: Sketch the angle and find the reference angle θ′ for each angle θ given.
a)=
θ 320°
d) θ =
7π
9
7π
6
b) θ = –150°
c) θ = −
e) θ = 5.29
f) θ = 64°
*If an angle is negative, you can ignore the negative sign and proceed as normal.
199
Finding Reference Angles for Angles Greater than 360° (2π) or less than –360° (–2π)
1. Find a positive angle α less than 360° or 2π that is coterminal with the given angle.
2. Draw α in standard position.
3. Use the drawing to find the reference angle for the given angle. The positive acute angle formed by the terminal side of α and the
x-axis is the reference angle.
Example 2: Sketch the angle and find the reference angle θ ´ for each angle θ given.
8π
3
a) θ = 580°
b) θ =
c) θ = –640°
d) θ = −
13π
6
Example 3: Find the reference angle θ ´ for each angle θ given.
a)
b)
200
How to find a reference angle (θ´) when given Positive θ
Quadrant 1
Quadrant 2
Radians: __________________
Radians: ____________________
Degrees: __________________
Degrees: ____________________
Quadrant 3
Quadrant 4
Radians: __________________
Radians: ____________________
Degrees: __________________
Degrees: ____________________
201
202
Pre-Calculus
Trig Day 5 Worksheet
Name: _____________________________________
For numbers 1 & 2, a point on the terminal side if angle θ is given. Find the exact value of each of the six trigonometric
functions of θ.
1. (2, 3)
2. (–4, 3)
For numbers 3 – 5, find the exact value of each of the remaining trigonometric functions of θ.
3. sin θ =
5
, θ in quadrant II
13
5. tan θ =
5
, cos θ < 0
12
2
4. tan θ = − , sin θ > 0
3
For numbers 6 – 11, find the reference angle for each angle.
6. 170°
9.
11π
4
5π
6
7. 455°
8.
10. –359°
11. 4.7
203
204
Pre-Calculus
Trig Day 5 Homework
Name: _____________________________________
For numbers 1 & 2, a point on the terminal side if angle θ is given. Find the exact value of each of the six trigonometric
functions of θ.
1. (–2, –5)
2. (–12, 5)
For numbers 3 – 5, find the exact value of each of the remaining trigonometric functions of θ.
3
3. cos θ = − , θ in quadrant III
5
4. cos θ =
8
, 270° < θ < 360°
17
5. csc θ = –4, tan θ > 0
For numbers 6 – 11, find the reference angle for each angle.
6. 160°
9.
5π
7
7. 705°
10. −
13π
4
8. 351°
11. 2.65
205
206
Trig Day 6 Notes
Evaluating Trigonometric Functions Using Reference Angles
The values of the trigonometric functions of a given angle, θ are the same as the values of the trigonometric functions of the reference
angle θ ´, except possibly for the sign. A function value of the acute reference angle, θ ´, is always positive. However, the same
function value for θ may be positive or negative.
A Procedure for Using Reference Angles to Evaluate Trigonometric Functions
The value of a trigonometric function of any angle θ is found as follows:
1. Find the associated reference angle, θ ´, and the function value for θ ´.
S
A
T
C
2. Use the quadrant in which θ lies to prefix the appropriate sign to the function value in step 1.
Example 1: Express each of the following in terms of a reference angle.
a) sin 128°
b) cos 128°
c) sin (–37°)
d) cos 500°
e) sec 280°
f) csc 115°
g) tan (–140°)
h) cot
5π
6
i) sec
7π
4
207
208
Pre-Calculus
Trig Day 6 Homework
Name: _____________________________________
For numbers 1 – 6, express each of the following in terms of a reference angle.
1. sin 205°
2. cos 991°
4. tan 120°
5. csc
7. sin 447°
8. cos 147°
10. tan 622°
11. csc
3. sec (–172°)
5π
6
6. cot
7π
3
9. sec (–22°)
5π
3
12. cot
3π
4
For numbers 13 & 14, find one positive and one negative coterminal angle.
13. 415°
14.
19π
6
15. A sector of a circle has radius 9 cm and central angle 44˚. Find its arc length and area. Round answers to the nearest tenth.
16. A sector of a circle has arc length 22 cm and central angle 5.3 radians. Find its radius and area. Round answers to the nearest
tenth.
17. A sector of a circle has area 35 cm2 and central angle 27˚. Find its radius and arc length. Round answers to the nearest tenth.
209
18. Find the measures of the sides of the right triangle whose lengths are designated by a lowercase letter. Round answers to the
nearest whole number.
B
221 in.
a
34°
A
b
C
19. At a point on the ground 50 feet from the foot of a tree, the angle of elevation to the top of the tree is 53º. Find the height of the
tree.
20. From the top of a lighthouse 210 feet high, the angle of depression to a boat is 27º. Find the distance from the boat to the foot of
the lighthouse. The lighthouse was built at sea level.
For numbers 21 & 22, find the exact value of each of the remaining trigonometric functions of θ.
21. sin θ =
2
22. tan θ = − , sin θ > 0
3
5
, θ in quadrant II
13
For numbers 23 – 25, evaluate the trigonometric function at the quadrantal angle, or state that the expression is undefined.
23. tan π
24. csc π
25. cot
π
2
For numbers 26 – 28, find the reference angle for each angle.
26. 170°
27. 455°
28.
5π
6
210
Trig Day 7 Notes
Introduction to the Unit Circle
Example 1: Use reference angles or the unit circle to find the exact value of each of the following trigonometric functions.
a) sin 135°
b) cos
4π
3
 π
c) cot  − 
 3
d) tan
14π
3
e) sec840
 7π 
f) csc  −

 4 
211
g) sec ( −315 )
h) tan 720˚
i) cot ( −7π )
Example 2: Find the exact value of each expression. Write the answer as a single fraction. Do not use a calculator.
a) sin
b) sin
π
3
π
4
cos π − cos
cos 0 − sin
π
3
π
6
sin
3π
2
cos π
212
Pre-Calculus
Trig Day 7 Homework
Name: _____________________________________
For numbers 1 – 9, use reference angles or the unit circle to find the exact value of each expression. Do not use a calculator.
1. cos 225°
 17π 
3. sin  −

 3 
2. tan 405°
7π
6
4. cot
7π
4
5. csc
7. tan
9π
4
 35π 
8. sin  −

 6 
6. tan 210°
9. sec 495°
For numbers 10 & 11, find the exact value of each expression. Write the answer as a single fraction. Do not use a calculator.
10. sin
11π
5π
11π
5π
cos
sin
+ cos
4
6
4
6
11. sin
3π
 5π 
 15π 
tan −
 − cos −

4
2

 3 

For numbers 12 & 13, find the exact value of each of the remaining trigonometric functions of θ.
12. tan θ =
5
, cos θ < 0
12
13. csc θ = −
25
, π ≤ θ ≤ 2π
7
213
For numbers 14 – 16, find the reference angle for each angle.
14.
11π
4
15. –359°
16. 4.7
17. From the top of a lighthouse 160 feet high, the angle of depression of a boat out at sea is an angle of 24°. Find to the nearest foot
the distance from the boat to the foot of the lighthouse, the foot of the lighthouse being at sea level.
For numbers 18 & 19, use the given point on the terminal side of an angle θ in standard position to evaluate the six trigonometric
functions.
18. (–12, 5)
19. (15, –8)
For numbers 20 – 22, name the quadrant described in the following:
20. tan θ > 0 and cos θ < 0
21. sin θ < 0 and sec θ > 0
22. cot θ < 0 and csc θ > 0
23. A sector of a circle has central angle 70° and arc length 7.5 cm. Find its area to the nearest square centimeter.
24. A sector of a circle has radius 5 cm and central angle 137°. Find it approximate arc length and area. Round answers to the nearest
tenth.
For numbers 25 – 30, express each of the following in terms of a reference angle.
25. sin 447°
26. cos 147°
28. tan 622°
29. csc
5π
3
27. sec (–22°)
30. cot
3π
4
214
Trig Day 8 Notes
Inverse Trig Functions
*Before, we evaluated trigonometric functions from any angle. (We found side lengths)
*Now, we can use inverse trigonometric functions to FIND the angle!!!
(When we are given a value of a trigonometric function or the side lengths)
In the following definitions, a denotes the given value
* If –1 ≤ a ≤ 1, then the inverse sine of a is sin-1a = θ, where sinθ = a and –90° ≤ θ ≤ 90°
–1 ≤ sin θ ≤ 1
θ
* If –1 ≤ a ≤ 1, then the inverse cosine of a is cos-1a = θ, where cos θ = a and 0° ≤ θ ≤ 180°
–1 ≤ cos θ ≤ 1
θ
* If –1 ≤ a ≤ 1, then the inverse tangent of a is tan-1a = θ, where tan θ = a and –90° < θ < 90°
–∞ < tan θ < + ∞
θ
Note: There are restrictions on which quadrant θ can be in to keep it a function. For example:
215
Evaluating Inverse Trigonometric Functions
*Basically, you need to think of where the given value is on the unit circle to determine the angle measure.
Example 1: Find the exact value of the expression, your answer must be in radians.
a) sin-1
2
2
b) cos-1 3
c) tan-1 1
d) tan-1 − 3
Example 2:
12 

a) Find tan  Cos −1  without a calculator.
13 

b) Find sec (Sin –1 (–0.5)) without a calculator.
Example 3: Find to the nearest degree the measure of the angle of elevation of the sun when a vertical post 15 feet high casts a
shadow 20 feet long.
Example 4: After takeoff, a plane flies in a straight line for a distance of 4000 feet in order to gain an altitude of 800 feet. Find to the
nearest degree the number of degrees contained in the angle in which the rising plane makes with the ground.
Example 5: Use a calculator to find the value of the acute angle θ (a) in radians, rounded to three decimal places, (b) in degrees,
rounded to the nearest degree.
sin θ = 0.4713
216
Pre-Calculus
Trig Day 8 Worksheet
Name: _____________________________________
For numbers 1 – 4, find the exact value of each expression, your answer must be in radians.
1. sin −1 0
 1
2. sin −1  − 
 2
3. cos −1
2
2
4. tan −1 1
For numbers 5 – 7, use a calculator to find the value of each expression in radians, rounded to two decimal places.
5. sin −1 0.3
6. sin −1 (− 0.625)
(
7. tan −1 − 473
)
For numbers 8 – 10, use a sketch to find the exact value of each expression.
7 

8. sin  tan −1

24




3 
9. csc cos −1  −

 2 




 3 
10. sin  tan −1  − 
 4 

For numbers 11 and 12, use a calculator to find the value of the acute angle θ in radians, rounded to three decimal places.
11. cos θ = 0.4112
12. tan θ = 0.4169
For numbers 18 and 19, use a calculator to find the value of the acute angle θ to the nearest degree.
13. sin θ = 0.2974
14. tan θ = 4.6252
15. A 20-foot ladder leans against a wall so that the base of the ladder is 8 feet from the base of the building. What angle does the
ladder make with the ground?
16. A telephone pole is 60 feet tall. A guy wire 75 feet long is attached from the ground to the top of the pole. Find the angle between
the wire and the pole to the nearest degree.
217
218
Pre-Calculus
Trig Day 8 Homework
Name: _____________________________________
For numbers 1 – 5, find the exact value of each expression, your answers must be in radians.
1. sin −1
1
2

3 
4. cos −1  −
 2 


2. sin −1
2
2
3. cos −1
3
2
( )
5. tan −1 − 3
For numbers 6 – 8, use a calculator to find the value of each expression in radians, rounded to two decimal places.
6. sin −1 0.47
7. cos −1
5
7
8. cos −1
4
9
For numbers 9 – 11, use a sketch to find the exact value of each expression.
4

9. cos sin −1 
5

5

10. tan cos −1 
13 


 1 
11. tan cos −1  − 
 4 

For numbers 12 and 13, use a calculator to find the value of the acute angle θ to the nearest degree.
12. cos θ = 0.8771
13. tan θ = 26.0307
For numbers 14 and 15, use a calculator to find the value of the acute angle θ in radians, rounded to three decimal places.
14. sin θ = 0.9499
15. tan θ = 0.5117
16. The Washington Monument is 555 feet high. If you are standing one quarter of a mile, or 1320 feet, from the base of the
monument and looking to the top, find the angle of elevation to the nearest degree.
17. A 50-meter vertical tower is braced with a cable secured at the top of the tower and tied 30 meters from the base. What angle does
the cable form with the ground?
219
For numbers 18 – 23, use reference angles or the unit circle to find the exact value of each expression. Do not use a calculator.
18. tan 210°
19. tan
9π
4
 35π 
20. sin  −

 6 
21. sec 495°
22. sec
5π
3
23. csc
3π
4
220
Pre-Calculus
Trig Unit 1 MC Packet
Name: _____________________________________
Day 1:
1. Which pair of angle measures represents coterminal angles?
a) 20° and 70°
b) 80° and 110°
c) 35° and –325°
d) 50° and –50°
e) 95° and –175°
2. Convert –60° to radians.
a) −
π
3
radians
3. Convert
b) −
π
6
radians
c)
π
6
radians
d)
π
3
radians
e) 3π radians
13π
radians to degrees.
6
a) 180°
b) 195°
c) 390°
d) 450°
e) 900°
c) –150°
d) 150°
e) 210°
d) 450° and –270°
e) 270° and –270°
4. Which is the angle of rotation shown below?
a) –330°
b) –210°
5. Which two angles are both coterminal with θ = 90°?
a) 270° and –90°
b) 270° and 450°
c) 0° and 180°
6. You start at 85° on the unit circle you make 7.25 complete rotations counterclockwise, what is the measure of the angle where you
land?
a) –175°
b) –5°
c) 0°
d) 5°
e) 175°
7. You start at 170° on the unit circle you make 6.35 complete rotations clockwise, what is the measure of the angle where you land?
a) 44°
b) 106°
c) 135°
d) 205°
e) 296°
c) 122°
d) 157°
e) 360°
c) 73°
d) 180°
e) 270°
d) 90°
e) 1170°
d) 90°
e) 630°
8. Which angle measure is closest to 1.8 radians?
a) 90°
b) 103°
9. Which angle measure is closest to
a) 0°
b) 67°
4
π
radians?
10. Give the angle measure represented by 3.25 rotations clockwise.
a) –1170°
b) –90°
c) 0°
11. Give the angle measure represented by 1.75 rotations counterclockwise.
a) –630°
b) –90°
c) 0°
221
12. Identify all coterminal angles between –360° and 360° for the angle –420°.
a) –60° and 300°
b) –30° and 330°
c) 30° and –330°
d) 60° and –300°
e) –60° and 330°
13. Change 1400° to radian measure in terms of π.
a)
35π
9
b)
14. Change
a) 66.6°
70π
9
c)
140π
9
d)
210π
9
e) None of these
29π
radians to degree measure.
37
b) 141.1°
c) 167.6°
d) 474.8°
e) 5220°
Day 2:
15. Find the arc length of sector if the central angle measures 90° and the diameter of the circle is 40 ft.
a) 10 ft
b) 31 ft
c) 35 ft
d) 62 ft
e) 93 ft
16. Find the arc length of sector if the central angle measures 240° and the diameter of the circle is 100 ft.
a) 209 ft
b) 418 ft
c) 471 ft
d) 623 ft
e) 942 ft
17. Find the degree measure of the central angle associated with an arc that is 21 centimeters long in a circle with a radius of 4
centimeters.
a) 10.9°
b) 59.2°
c) 68.6°
d) 183.5°
e) 300.8°
18. Find the area of a sector if the central angle measures 105° and the radius of the circle is 4.2 meters.
a) 7.7 m2
b) 16.2 m2
c) 32.3 m2
d) 847.2 m2
e) 926.1 m2
d) 18.12
e) 20.14
Day 3:
19. Find the value of x to the nearest hundredth of a foot.
x ft
16 ft
28°
a) 7.51
b) 8.51
c) 14.13
20. What is the cotangent of an angle if the sine of the angle is
a)
3
5
b)
3
4
c)
3
?
5
4
5
d)
5
4
e)
4
3
e)
12
5
21. The sides of right ∆ABC have lengths 10 ft, 24 ft, and 26 ft. Which could be the cosine of angle B?
a)
5
13
b)
5
12
c)
13
5
d)
13
12
22. A carpenter is making a ramp for a ski jump. The ramp is made from a 25 ft board titled at a 30° angle from the ground.
Approximately how far off the ground is the ramp?
a) 9.8 ft
b) 10.2 ft
c) 11.6 ft
d) 12.1 ft
e) 12.5 ft
222
23. A camper creates a make-shift tent by leaning a 15 ft branch against a tree and draping a tarp over it. The branch makes a 15°
angle with the ground. Approximately how tall is the tent at its highest point?
a) 3.89 ft
b) 4.02 ft
c) 9.63 ft
d) 14.49 ft
e) 15.56 ft
For numbers 24 & 25, use the figure. The angle of elevation from the end of the shadow to the top of the building is 63°and the
distance is 220 feet.
24. Find the height of the building to the nearest foot.
a) 100 ft
b) 112 ft
c) 196 ft
d) 274 ft
e) 432 ft
c) 196 ft
d) 274 ft
e) 432 ft
25. Find the length of the shadow to the nearest foot.
a) 100 ft
b) 112 ft
Day 4:
26. P(2, –8) is a point on the terminal side of θ in standard position. Find the exact value of cos θ.
a)
17
17
b)
2 17
17
c)
4 17
17
d)
10
5
e)
4 10
5
27. On the unit circle, in which quadrants is sin θ positive?
a) I and II
b) I and III
c) II and IV
d) III and IV
e) II and III
28. The point (3, –4) is on the terminal side of θ in standard position. Which is the sine of θ?
a) −
4
5
b) −
3
4
c) −
3
5
d)
3
5
e)
4
5
29. The point (–4, 6) is on the terminal side of θ in standard position. Which is the cosine of θ?
a) −
2 13
13
b) −
2
13
c)
2
13
d)
2 13
13
e) 13
30. Find the value of csc θ for angle θ in standard position if the point at (5, –2) lies on its terminal side.
a) −
29
2
b) −
2 29
29
c)
29
5
d)
2 29
5
e)
5 29
29
31. Find the value of sec θ for angle θ in standard position if the point at (–2, –4) lies on its terminal side.
a) − 5
b) −
5
2
c) −
5
3
d)
5
2
32. Suppose θ is an angle in standard position whose terminal side lies in Quadrant II. If sin θ =
a) −
13
5
b) −
12
5
c) −
5
13
d)
5
13
e) 5
12
, find the value of sec θ.
13
e)
13
12
223
3
1
33. The sine of θ is and the cosine of θ is −
. Which is the cotangent of θ?
2
2
a) − 3
3
2
b) −
c)
34. The tangent of θ is –1 and the cosine of θ is −
a) − 2
c)
t
R
6
3
b)
e) 2 3
2
3
d)
2
2
e) 2
S
35. Find the value of the cosecant for ÐR.
a)
d) 3
2
. Which is the cosecant of θ?
2
2
2
b) −
3
2
6
2
2
T
c)
15
3
d)
15
2
e)
10
2
c)
1
tan θ
d)
1
sec θ
e)
1
csc θ
36. Which of the following is equal to sec θ?
a)
1
sin θ
b)
1
cos θ
37. If cot θ = 0.85, find tan θ.
a) 0.588
b) 0.85
c) 1.12
d) 1.176
38. What is the value of csc θ? (Picture is not to scale)
e) 1.7
13
θ
12
5
a)
5
13
b)
5
12
c)
12
13
d)
13
12
e)
13
5
Day 5:
39. Which angle does NOT have a reference angle with a measure of 15°?
a) θ = –345°
b) θ = 255°
c) θ = 165°
d) θ = 195°
e) θ = 345°
c) 25°
d) 35°
e) 65°
d) 34°
e) 56°
40. What is the reference angle for θ = 165°?
a) 0°
b) 15°
41. Find the measure of the reference angle for 1046°.
a) –56°
b) –54°
c) –34°
224
Day 7:
42. Find cot (–180°).
a) –1
b) 0
c) 0.5
d) 1
e) undefined
c) 0
d)
2 3
3
e) 2
c) 0
d)
3
3
e) 3
43. Find the exact value of sec 300°.
a) –2
b) −
2 3
3
44. Find the exact value of tan 240°.
a) − 3
b) −
3
3
Day 8:
45. A road rises vertically 45 feet over a horizontal distance of 750 feet. What is the angle of elevation of the road? Round to the
nearest tenth of a degree.
a) 0.6°
b) 3.4°
c) 16.7°
d) 37.8°
e) 86.6°
46. A 23-ft ladder is leaning against a building and the base of the ladder is 12 ft from the building. Which is the closest
approximation of the angle the ladder makes with the ground?
a) 16°
47. Evaluate: Cos–1 −
a) 30°
b) 28°
c) 31°
d) 52°
e) 59°
c) 60°
d) 135°
e) 315°
2
.
2
b) 45°
48. Charles is trying to get his kite out of a tree. The kite is 12 feet higher than he can reach. If he stands 10 feet from the tree and can
throw a ball in a straight line, at what angle should he throw it in order to hit the kite?
a) 33.6°
b) 39.8°
c) 50.2°
d) 56.4°
e) 62.3°
 2
49. Which set of expressions, where n is an integer, gives all possible values of sin −1 
 ?
 2 
a) 2πn
b)
π
4
+ 2π n
c)
π
3
+ 2π n
d)
3π
+ 2π n
4
e)
7π
+ 2π n
4
e)
5π
+ 2π n
3

3 
?
50. Which set of expressions, where n is an integer, gives all possible values of cos −1  −
 2 


a) −
7π
+ 2π n
6
b)
π
6
+ 2π n
c)
5π
+ 2π n
6
d) π + 2πn
225
226
Pre-Calculus
Trig Unit 1 Review Sheet
Name: _____________________________________
1. Find the radian measure of the central angle of a circle of radius 6 centimeters that intercepts an arc of length 27 centimeters.
For numbers 2 – 4, convert each angle in degrees to radians. Express your answer in terms of π.
2. 15°
3. 120°
4. 315°
For numbers 5 – 7, convert each angle in radians to degrees.
5.
5π
3
6.
7π
5
7. −
5π
6
For numbers 8 – 12, draw each angle in standard position.
8.
5π
6
11. 190°
9. −
2π
3
10.
8π
3
15.
13π
4
12. –135°
For numbers 13 – 17, find one positive and one negative angle that is coterminal with the given angle.
13. 400°
16.
31π
6
14. –445°
17. −
8π
3
227
18. Find the length of the arc on a circle of radius 10 feet intercepted by a 135° central angle. Express arc length in terms of π. Then
round your answer to two decimal places.
19. Use the triangle to find each of the six trigonometric functions of θ.
B
5
θ
A
8
C
20. Find the exact value of the expression cos
2π
2π
sec
. Do not use a calculator.
9
9
For numbers 21 – 23, find a cofunction with the same value as the given expression.
21. sin 70º
23. cos
22. sec 38º
π
3
For numbers 24 – 26, find the measure of the side of the right triangle whose length is designated by a lowercase letter. Round
answers to the nearest whole number.
24.
B
B
25.
26.
B
c
20 cm
a
50 in.
61°
23°
A
a
A
100 mm
C
48°
C
C
A
27. When a six-foot pole casts a four-foot shadow, what is the angle of elevation of the sun? Round to the nearest whole degree.
For numbers 28 & 29, a point on the terminal side of angle θ is given. Find the exact value of each of the six trigonometric functions
of θ, or state that the function is undefined.
28. (–1, –5)
29. (0, –1)
228
For numbers 30 and 31, name the quadrant described in the following:
30. tan θ > 0 and sec θ > 0
31. tan θ > 0 and cos θ < 0
For numbers 32 and 33, find the exact value of each of the remaining trigonometric functions of θ.
32. cos θ =
2
, sin θ < 0
3
33. cot θ = 3, cos θ < 0
For numbers 34 – 36, find the reference angle for each angle.
34. 265º
35.
5π
8
36. –410º
For numbers 37 – 45, find the exact value of each expression.
37. sin 240º
40. cos
11π
6
 π
43. sin  − 
 3
7π
4
38. tan 120º
39. sec
41. cot(–210º)
 2π 
42. csc −

 3 
44. sin 495º
45. sin
22π
3
229
For numbers 46 – 55, find the exact value of each expression. Express your answer in terms of radians. Leave your answers in π
form.
46. sin −1 1
47. cos −1 1
48. tan −1 1

3 
49. sin −1  −
 2 



3 
50. tan −1  −
 3 



2 
51. cos sin −1

2 



3  
52. tan cos −1  −
 2 




3

53. cos tan −1 
4


3 
54. csc tan −1

3 


 1 
55. sin  tan −1  −  
 3 

For numbers 56 – 58, express each of the following in terms of a reference angle.
56. sin 597 º
57. sec
5π
3
58. cot (–300 º)
230