Download Honors Pre-calculus Chapter 6 Assignments/Schedule - Spring

HONORS PRE-CALCULUS
CHAPTER 6 ASSIGNMENTS/SCHEDULE
BE SURE TO READ EACH SECTION BEFORE YOU DO THE PROBLEMS ASSIGNED!
DATE
SECTION
ASSIGNMENT
1.) ________
6.1
P. 436, # 4, 8, 12, 16, 36, 44, 45, 46
2.) ________
6.1
P. 437-438, # 6, 19, 20, 21, 23, 27, 28, 35, 37
3.)________
6.1
P. 437-438, # 29, 30 AND [38, 43]
4.) ________
6.2
P. 443; # [4, 16] EVENS AND [24, 28] EVENS
5.) ________
6.2
P. 443; # [32, 48] MULTIPLES OF 4
6.) ________
6.1-6.2
REVIEW WORKSHEET
7.) ________
QUIZ 6.1 – 6.2
QUIZ
8.) ________
6.5
P. 478: #[8,64] MULTIPLES OF 4
9.) ________
6.5
P. 478; # 9, 17, 25, 29, 47, 53, 61 [73, 87] ODDS
10.) ________
6.5
P. 479; # [91, 97] ODDS
11.) ________
6.1-6.5
REVIEW P. 482 # [6, 36] MULT OF 6 AND P. 484 # [99-112] MULTIPLES OF 3
12.) ________
6.1-6.5
TEST
Honors Pre-Calculus – Chapter 6
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Honors Pre-Calculus – Chapter 6
[2]
6.1 LAW OF SINES
Student will be able to:
•use the law of sines to solve oblique triangles with AAS, ASA or SSA
the law of sines to solve real-life problems
LAW OF SINES
Oblique Triangle:
To solve you need to know one of the four cases of given information:
1.) Two angles and any side (AAS or ASA)
2.) Two sides and an angle opposite one of them (SSA)
3.) Three sides (SSS)
4.) Two sides and there included angle (SAS)
Law of Sines
If ABC is a triangle with sides a, b, and c, then
a
b
c


sin A sin B sin C
A is acute
A is obtuse
NOTE: The Law of Sines can be written in reciprocal form:
Honors Pre-Calculus – Chapter 6
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•find the area of oblique triangles •use
⬄ (AAS) Given: C  102.3 , B  28.7 , and b  27.4 feet. Find the remaining angle and sides.
⬄ (ASA) Given: A pole tilts toward the sun at an 8 angle from the vertical, and it casts a 22-foot shadow. The angle
of elevation from the tip of the shadow to the top of the pole is 43 . How tall is the pole?
Honors Pre-Calculus – Chapter 6
[4]
Law of Sines – The Ambiguous Case (SSA)
3 possibilities exist!!!!!!!
1. No such triangle exists
2. One triangle exists
3. Two distinct triangles exist
⬄ Given a  22 , b  12 , and A  42 .
Honors Pre-Calculus – Chapter 6
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⬄ Given a  15 , b  25 , and A  85 .
Honors Pre-Calculus – Chapter 6
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⬄ Given a  12 , b  31, and A  20.5
Honors Pre-Calculus – Chapter 6
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Area of an Oblique Triangle
1
AREA  bc sin A
2
A
1
ab sin C
2
A
1
ac sin B
2
**In other words, the area equals one-half the product of 2 sides and the sine of their included angle.
⬄ Find the area of a triangular lot with two sides of lengths 90 meters and 52 meters, and an included angle of 102
degrees.
Honors Pre-Calculus – Chapter 6
[8]
6.1 LAW OF SINES PRACTICE #1
1.)
a  12 , mB  70 , mC  15
2.)
a  12 , b  5 , mA  110
3.)
a  8 , mA  60 , mC  40
Honors Pre-Calculus – Chapter 6
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4.)
a  5 , c  4 , mA  65
5.)
b  6 , mA  44 , mB  68
6.)
a  7 , mA  37 , mB  76
Honors Pre-Calculus – Chapter 6
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7.)
a  9 , b  9 , mC  20
8.) A ship is sighted from two radar stations 43 km apart. The angle between the line segment joining the two stations
and the radar beam of the first station is 37 . The angle between the line segment joining the two stations and the
beam from the second station is 113 . How far is the ship from the second station?
Honors Pre-Calculus – Chapter 6
[11]
6.2 THE LAW OF COSINES
Student will be able to:
•use the law of cosines to solve oblique triangles(SSS or SAS) •use the law of cosines to model and solve realworld problems
•use Heron’s formula to find the area of a triangle
There are two cases that require the Law of Cosines for oblique triangles, SSS and SAS:
Standard Form
Alternative Form
a  b  c  2bc cos A
b2  c 2  a 2
cos A 
2bc
b 2  a 2  c 2  2ac cos B
cos B 
a 2  c 2  b2
2ac
c 2  a 2  b 2  2ab cos C
cos C 
a 2  b2  c 2
2ab
2
2
2
⬄ Given a  8 ft, b  19 ft, and c  14 ft, find all 3 angles.
Honors Pre-Calculus – Chapter 6
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⬄Given b  15 cm, c  10 cm, and A  115 , find the remaining side and angles.
Application of the Law of Cosines.
⬄The pitcher’s mound on a women’s softball field is 43 feet from home plate and the distance between the bases is 60
feet. How far is the pitcher’s mound from first base? (The mound is not halfway between home and second base.)
Honors Pre-Calculus – Chapter 6
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Heron’s Area Formula (For SSS)
Area  s  s  a  s  b  s  c  where s 
abc
.
2
⬄ Find the area of the triangle with sides 12 feet, 15feet, and 9 feet.
Honors Pre-Calculus – Chapter 6
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LAW OF COSINES (6.2 PRACTICE)
Solve each triangle, rounding to the nearest tenth.
1. a  16
b  20
mB  40
2. a  10
b  15
c  12
3. a  42
c  60
mB  58
Honors Pre-Calculus – Chapter 6
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4. mA  60
5. a  7
mB  72
b  12
6. mA  43
c9
c  15
b  23
Honors Pre-Calculus – Chapter 6
c  26
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7. a  16
8. c  15.6
mA  23
a  12.9
Honors Pre-Calculus – Chapter 6
mB  87
b  18.4
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6.1-6.2 Review
1.) Given A  24 , B  68 , and a  12.2 , find c.
2.) Find the obtuse angle (if one exists) for a triangle with sides a  27.52 , b  31.11, and c  24.98 .
3.) Find all parts of the triangle given A  24 , a  11.2 , and b  13.4 .
Honors Pre-Calculus – Chapter 6
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4.) Find all parts of the triangle given C  123 , a  41 , and b  57 .
5.) A triangular parcel of land has borders of lengths 60 meters, 70 meters, and 82 meters. Find the area of the piece of
land.
Honors Pre-Calculus – Chapter 6
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6.5 Trigonometric From of a Complex Number
Recall the complex plane
Horizontal axis
Vertical axis
Graph:
A. -2-3i
B. 4+i
C. 2i
Absolute Value (distance between the origin and the point (a,b) of a complex number z = a + bi is given by
a  bi  a2  b2
Evaluate |4 – 2i|
The Trigonometric Form of a complex number z = a + bi is z  r (cos   i sin  ) , where a  r cos , b  r sin  ,
r  a 2  b2 and tan  
b
. The number r is the modulus of z, and  is the argument of z (NOTE: This form is also
a
called the polar form)
Write
z  2  2i 3 in trigonometric form.
Honors Pre-Calculus – Chapter 6
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Multiplication and Division of Complex Numbers
Let z1  r1 (cos 1  i sin 1 ) and z2  r2 (cos 2  i sin 2 ) be complex numbers. THEN:
z1 z2  r1r2 cos 1   2   i sin 1   2  
You….
z1 r1
 cos 1   2   i sin 1   2   , z2  0
z2 r2 
You …
Ex3: Given z1  2(cos
2
2
11
11
 i sin ) and z2  8(cos
 i sin
)
3
3
6
6
Find: z1 z2 
Find:
z1

z2
Honors Pre-Calculus – Chapter 6
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Powers of a Complex Number
z n  r  cos   i sin   
DeMoivre’s Theorem – If z  r (cos   i sin  ) is a complex integer, then
 r n  cos n  i sin n 
n
2
2 

 i sin ) 
Ex4:  2(cos
3
3 

12
Definition of the nth Root of a Complex Number - The complex number u  a  bi is an nth root of the complex
number z if z  u n   a  bi 
n
Finding the nth root of a Complex Number – For a positive integer n, the complex number z  r (cos   i sin  ) has
exactly n distinct nth roots given by
n
  2 k
  2 k 

r  cos
 i sin
 where k = 0, 1, 2, …..n-1.
n
n


Ex. Find the sixth roots of 1
Honors Pre-Calculus – Chapter 6
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Ex. Fin the three cube roots of z  2  2i
Honors Pre-Calculus – Chapter 6
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