Download SD 9-12 Misc

SD 9-12 Misc. - all other types of questions
03/20/2009
Student Name:
___________________________
Class:
___________________________
Date:
___________________________
Instructions:
Read each question carefully and select the correct answer.
1.
Find the cosine of angle Z in the triangle
below. Rationalize the denominator if
necessary.
2.
Julio is 6 feet, 2 inches tall. At 4 P.M., he
studies his shadow and wants to know how
long it is. Determine the length of the
shadow if the angle from the top of Julio's
shadow to the top of his head is 26º . Round
your answer to the nearest inch.
A.
B.
A.
B.
C.
D.
C.
152 inches
82 inches
169 inches
36 inches
D.
3.
Simplify.
A.
B.
C.
D.
4.
Find 25% of 75.
A.
B.
C.
D.
SD 9-12 Miscellaneous Math Topics – Test, Answer Key, Study Guide
41
- 41
2
-2
18.75
3
300
1,875
Page 1 of 36
5.
Multiply.
9.
Find the sine, cosine, and tangent of angle
DFE.
(- 45)(- 21)
A.
B.
C.
D.
6.
35º
- 395º
55º
145º
Ruben is saving money to buy a new car.
Today, he deposited money into a savings
account at ABC Bank which pays 5.4%
interest, compounded monthly. The car that
Ruben wants will cost
$10,250.00 with tax and license. According
to his calculations, he will have enough
money in two years. How much money did
Ruben deposit into the bank?
A.
B.
C.
D.
8.
A.
B.
C.
D.
Find the reference angle for 395º .
A.
B.
C.
D.
7.
135
- 135
945
- 945
10.
12.
$33.72
$349.84
$337.20
$375.95
-150 ÷ 25 =
A.
B.
C.
D.
$11,416.22
$9712.36
$9202.91
$2901.01
Which choice best describes the following
group of numbers?
Your credit card currently has a balance of
$4,214.96. The annual interest rate on your
card is 8%. How much interest will accrue
in one year?
A.
B.
C.
D.
11.
12/13, 5/13, 5/12
5/13, 12/13, 5/12
13/12, 13/5, 12/5
13/5, 13/12, 12/5
-3750
-6
3750
6
Compare. Use <, >, or =.
- 13 + 12 _______________ - 12 + 13
2, -3, 5, 0
A.
B.
C.
D.
whole numbers
integers
irrational numbers
natural numbers
SD 9-12 Miscellaneous Math Topics – Test, Answer Key, Study Guide
A.
B.
C.
>
<
=
Page 2 of 36
13.
Triangle ABC is a right triangle. Which
statement applies to the triangle?
14.
A 23 meter ladder is leaning against a
building. The angle formed by the ground
and the bottom of the ladder is 51º . Find the
height at which the top of the ladder reaches
the building to the nearest hundredth.
A.
B.
A.
B.
C.
D.
17.87 meters
35.72 meters
74.06 meters
161.08 meters
C.
D.
SD 9-12 Miscellaneous Math Topics – Test, Answer Key, Study Guide
Page 3 of 36
SD 9-12 Misc. - all other types of questions
Answer Key
03/20/2009
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
D
A
C
A
C
A
C
B
B
C
B
B
A
A
Trigonometric Ratios - A
Solving Right Triangles - A
Absolute Value: Simplify
Percent of a Number
Multiplying Integers
Functions: Trigonometric
Compound Interest
Evaluating Solutions - A
Trigonometric Identities
Credit
Dividing Integers
Integers: Multiple-step Computation
Trigonometric Ratios - B
Solving Right Triangles - B
SD 9-12 Miscellaneous Math Topics – Test, Answer Key, Study Guide
Page 4 of 36
Study Guide
SD 9-12 Misc. - all other types of questions
03/20/2009
Trigonometric Ratios - A
This study guide will focus on recognizing and evaluating the three trigonometric functions: sine, cosine, and
tangent. In order to properly utilize these functions, the student needs to be familiar with the parts of a right
triangle.
Important Terms:
• A right triangle is a polygon with three sides, where one of its angles is exactly 90º .
• A hypotenuse is the side opposite the right angle (the slant or longest side) in a right triangle.
• The two sides of a right triangle that are not the hypotenuse are called legs.
• The sum of the interior angles of a triangle is 180º . Since one angle of a right triangle is 90º , the other
two angles must each be less than 90º , which makes them acute.
Consider the triangle below. Using angle C as a reference, line segment AB is the side opposite the
angle, segment BC is the side adjacent to the angle, and segment AC is the hypotenuse.
Determining the Sine of an Angle:
SD 9-12 Miscellaneous Math Topics – Test, Answer Key, Study Guide
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Determining the Cosine of an Angle:
Determining the Tangent of an Angle:
Example 1: Find the sine of
T in the triangle below.
Step 1: Determine which components to use for the sine of an angle (opposite and hypotenuse).
Step 2: Substitute 8 for length of opposite and 10 for length of hypotenuse into the formula.
Step 3: Reduce.
Example 2: Find the cosine of
T in the triangle below.
Step 1: Determine which components to use for the cosine of an angle (adjacent and hypotenuse).
Answer:
SD 9-12 Miscellaneous Math Topics – Test, Answer Key, Study Guide
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Example 3: Find the tangent of
T in the triangle below.
Step 1: Determine which components to use for the tangent of an angle (opposite and adjacent).
Step 3: Rationalize the denominator, that is,
write an equivalent expression that does not contain any radicals in the denominator.
Answer:
The following is a helpful way to remember the trigonometric functions. It is pronounced "So"-"Ca""Tow-a".
To reinforce this skill have the student make several right triangles with different side lengths. An easy
way to do this is by taking an 8.5 by 11 piece of paper and folding it diagonally. Cut along the diagonal
fold to create two right triangles. Make a cut parallel to the hypotenuse of one of the right triangles to
make a smaller right triangle (see diagram below). Measure the legs and hypotenuse of the triangles
with a ruler and record the results. Have the student find the sine, cosine, and tangent of the triangles,
picking a different acute angle each time.
SD 9-12 Miscellaneous Math Topics – Test, Answer Key, Study Guide
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Solving Right Triangles - A
This study guide will focus on solving real world problems using right triangles that involve trigonometric
concepts. Real world problems are often difficult for students. It may be beneficial to confirm that the student
is comfortable with trigonometric functions and the right triangle outside of a real world context. Begin by
reviewing the definitions and calculation procedures for determining sine, cosine, and tangent with the student.
Calculator use is recommended.
Important Terms:
• A right triangle is a polygon with three sides, where one of its angles is exactly 90º .
• A hypotenuse is the side opposite the right angle (the slant or longest side) in a right triangle.
• The two sides of a right triangle that are not the hypotenuse are called legs.
• The sum of the interior angles of a triangle is 180º . Since one angle of a right triangle is 90º , the other
two angles must each be less than 90º , which makes them acute.
Consider the triangle below. Using angle C as a reference, line segment AB is the side opposite the
angle, segment BC is the side adjacent to the angle, and segment AC is the hypotenuse.
Determining the Sine of an Angle:
Determining the Cosine of an Angle:
SD 9-12 Miscellaneous Math Topics – Test, Answer Key, Study Guide
Page 8 of 36
Determining the Tangent of an Angle:
The following is a helpful way to remember the trigonometric functions. It is pronounced "So"-"Ca""Tow-a".
Example 1: Find the sine of
T in the triangle below.
Step 1: Determine which components to use for the sine of an angle (opposite and hypotenuse).
Step 2: Substitute 8 for length of opposite and 10 for length of hypotenuse into the formula.
Step 3: Reduce.
Right Triangles in Story Problems:
In story problems dealing with real world situations, the measure of the reference angle and the measure
of one side of the triangle are usually given. It is the job of the student to recognize which trigonometric
function is needed and then substitute the given information into the formula and solve for the missing
side. See the example below.
Example 2: In the beginning of her morning surf session, Li Ming paddled due west 100 feet from
shore, then the current carried her due north. At his point, Li Ming was at a 27º angle with the spot she
started on the shore. How far north did the current take her? Round your answer to the nearest tenth of
a foot.
SD 9-12 Miscellaneous Math Topics – Test, Answer Key, Study Guide
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Step 1: Determine which trigonometric function is needed to solve the problem. The only acute angle
given in the problem is 27º , so the trigonometric function must use that angle. The length of the
adjacent side is needed and the length of the opposite side is given, so the tangent function is the correct
function to use.
Step 2: Substitute 27º for the angle and 100 for the length of the opposite side.
Step 3: Multiply both sides of the equation by x to eliminate the denominator.
Step 4: Simplify.
Step 5: Divide both sides of the equation by tan 27º to isolate the x.
Step 6: Have the student use a calculator or a trigonometry table (generally found in the back of the text
book) to determine that the tangent of 27º is approximately 0.5095. Substitute this value into the
equation in place of tan 27º .
Step 7: Divide 100 by 0.5095 and round to the nearest tenth to get 196.3.
Answer: Li Ming was carried north 196.3 feet by the current.
Have the student research where right triangles are used in everyday situations. Resources that may
prove useful include magazines, newspapers, educational television shows, and the Internet.
Absolute Value: Simplify
The absolute value of any number is its distance from zero on a number line. Absolute value is always a
positive value because it represents a distance.
SD 9-12 Miscellaneous Math Topics – Test, Answer Key, Study Guide
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•The absolute value of -3 (written |- 3|) is 3, since it is 3 units away from zero.
•The absolute value of 4 (written |4|) is 4, since it is 4 units from zero.
The absolute value of a number is denoted with the following sign or symbol, | |. It works as a
grouping symbol and therefore falls in the category of parentheses for order of operations. To evaluate
an absolute value expression, perform the calculations inside the absolute value sign first, then take the
positive value and "drop" the absolute value sign. Continue calculations using order of operations. For
this skill, the result should be numeric.
A review of order of operations is provided below.
P = Parenthesis, E = Exponents, M = Multiplication, D = Division, A = Addition, S = Subtraction.
**Note: Evaluate multiplication and division from left to right (whichever comes first). Evaluate
addition and subtraction from left to right (whichever comes first).
Simplifying Absolute Value Expressions
Example 1: Simplify.
Step 1: Follow order of operations by multiplying first.
Step 2: Addition and subtraction are the remaining operations so perform the calculations from left to
right. 7 + 2 = 9.
Step 3: Subtract 9 - 5 = 4.
Step 4: Subtract 4 - 16 = -12.
Step 5: Take the absolute value of -12 which is 12.
Answer: 12
Example 2: Simplify.
SD 9-12 Miscellaneous Math Topics – Test, Answer Key, Study Guide
Page 11 of 36
Step 1:
Step 2:
Step 3:
Step 4:
Step 5:
Start inside the absolute value sign. Divide first. 8 ÷ 2 = 4.
Add. 4 + 7 = 11
The absolute value of 11 is 11.
Multiply. 2 11 = 22
Subtract. 24 - 22 = 2.
Answer: 2
A good way to reinforce the concept of absolute value is to ask the student if he or she is able to think of
how and when absolute values are used in everyday life (temperature, elevation, time A. D. or B. C.,
etc.).
Percent of a Number
Percent means "per one hundred." For example, if 7 out of 100 students ate pizza for lunch, then 7% (7 percent)
of the students ate pizza for lunch.
The student should understand how to determine the percent of a number.
Example: Find 25% of 48.
Step 1: Change the percent amount to a fraction (remember percent means "per one hundred").
Step 2: Multiply 25/100 by 48. Change 48 into a fraction by making its denominator 1. Multiply
numerator by numerator (25 x 48 = 1,200). Multiply denominator by denominator (100 x 1 = 100).
Step 3: Reduce the product to its lowest terms (1,200 ÷ 100 = 12).
Answer: 12
Multiplying Integers
Integers are the set of positive and negative whole numbers, including zero. Students should understand how
integers appear on a number line. Numbers to the right of 0 on a number line are positive and numbers to the
left of 0 are negative. The number -3 is a negative integer and the number 3 is a positive integer. The integer 0
is neutral.
To multiply and divide integers, follow these rules.
•The product of two positive integers is positive (Example: 9 x 4 = 36).
SD 9-12 Miscellaneous Math Topics – Test, Answer Key, Study Guide
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•The product of a positive integer and a negative integer is negative (Example: 9 x -4 = -36).
•The product of two negative integers is positive (Example: -9 x -4 = 36).
Notice from the above examples that when you multiply integers with the same sign, the answer is positive.
When multiplying integers with different signs, the answer is negative.
The order of operations must be followed when working with grouping symbols and/or multiple-step
operations. The order of operations are as follows:
1. Work inside grouping symbols
2. Multiply and divide from left to right
3. Add and subtract from left to right
Example 1: (3 x -5) + 7 = ?
(1) 3 x -5 = -15
(2) -15 + 7 = -8
Step 1: Using the order of operations, calculate within grouping symbols. 3 x -5 = -15, so we can replace the
parentheses with the value of -15.
Step 2: Perform the addition.
Answer: -8
Example 2: 8(-3 x 4) x (-5 + 7) = ?
(1) -3 x 4 = -12 and -5 + 7 = 2
(2) 8(-12) x 2 = ?
(3) 8(-12) = -96
(4) -96 x 2 = -192
Step 1:
Step 2:
Step 3:
Step 4:
Work inside grouping symbols.
Rewrite the problem with the new numbers.
Multiply from left to right, 8 x -12 = -96
Multiply -96 by 2 to get -192.
Answer is: -192
Functions: Trigonometric
Trigonometric functions set up relationships between the acute angles (less than 90º ) and ratios of the side
lengths of right triangles (any triangle with a 90º angle).
The trigonometric functions of A (one of the acute angles of a right triangle) in terms of the lengths of
the sides of the right triangle are below.
SD 9-12 Miscellaneous Math Topics – Test, Answer Key, Study Guide
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Consider triangle ABC:
In triangle ABC, side AB is the hypotenuse because it is opposite the right angle (the angle marked with
a box in its corner). If attention is focused on angle A (expressed A, BC is the opposite side and AC is
the adjacent side. Using the trigonometric functions above, we can determine numerical values of the
sine, cosine, tangent, cosecant, secant, and cotangent of A.
sin A = 4/5
cos A = 3/5
tan A = 4/3
csc A = 5/4
sec A = 5/3
cot A = 3/4
The trigonometric functions will also work for finding the sine, cosine, tangent, cosecant, secant, and
cotangent of B, but they will not work for the right angle, C. The values of sine and cosine for acute
angles will always be in the range from -1 to 1.
Converting from Degrees to Radians:
To convert from degrees to radians, we use the formula:
Example 1: Convert 320º to radians.
Solution: Substitute the degree of the angle into the formula for converting degrees to radians. Reduce
the fraction, if possible, to determine the final answer.
SD 9-12 Miscellaneous Math Topics – Test, Answer Key, Study Guide
Page 14 of 36
Answer:
Reference Angles
A reference angle is defined as the positive acute angle formed by the terminal side of an angle and the
x-axis, as long as the given angle is in standard position and the terminal side does not lie on either the
x-axis or the y-axis. An angle is in standard position if it is placed on a coordinate system with its vertex
at the origin. The four angles below are all in standard position.
Figure A: the reference angle = 25º
Since the angle in this figure is less than 90º , the reference angle is simply the degree of the angle.
Figure B: the reference angle = 180º - 113º = 67º
Since the angle in this figure is greater than 90º and smaller than 180º , the reference angle is the
difference between 180º and the degree of the angle.
Figure C: the reference angle = 235º - 180º = 55º
Since the angle in this figure is larger than 180º and smaller than 270º , the reference is the difference
between the degree of the angle and 180º .
Figure D: the reference angle = 360º - 345º = 15º
Since the angle in this figure is larger than 270º , the reference angle is the difference between 360º and
the degree of the angle.
Amplitude and Period
A periodic function is a function that repeats itself at regular intervals. The period of a function is the
length of the interval it takes for the function to repeat its pattern. y = sin x is a periodic function, its
graph is shown below.
The period of y = sin x is
SD 9-12 Miscellaneous Math Topics – Test, Answer Key, Study Guide
Page 15 of 36
The standard forms for the periodic functions of sine and cosine are:
y = a sin (kx + b)
y = a cos (kx + b)
In the standard forms, x and y remain variables, but a, b, and k are replaced by numbers. The amplitude
of the graph of the periodic functions of sine and cosine is the absolute value of a (|a|) and the period of
the functions is
The amplitude of a periodic function is half the maximum peak-to-peak value. The amplitude of y = sin
x is 1, because |1| = 1.
Here are two more examples.
Example 2: What is the amplitude of the graph of y = 3 sin (2x + 4)?
amplitude = |a|
a=3
|3| = 3
Solution: Since the amplitude of a periodic function is the absolute value of a, we need to determine the
value of a. By looking back at the standard form of the sine function, we can see that the 3 in
y = 3 sin (2x + 4) is in the same place as a in y = a sin (kx + b); therefore, a = 3. The amplitude of
y = 3 sin (2x + 4) is 3.
Example 3: What is the period of the graph of y = 5 cos (4x +6)?
Solution: By looking back at the standard form of the cosine function, we can see that 4 in
y = 5 cos (4x + 6) is in the same place as k in y = a cos (kx + b); therefore, k = 4. The period of
y = 5 cos (4x + 6) is
Inverse Trigonometric Functions
Each of the trigonometric functions has an inverse function. For example, the inverse sine of a variable
is called the arc sine of x; it is written arcsin x and is the angle for which the sine equals the value of x.
The other inverse trigonometric functions we will use are:
inverse cosine of x: arc cosine, written arccos x.
inverse tangent of x: arc tangent, written arctan x.
Example 4: Find arcsin (0.9877).
(1) arcsin (0.9877) = y
(2) sin y = 0.9877
(3) sin 81º = 0.9877
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(4) arcsin (0.9877) = 81º
Step 1: Set the inverse sine equal to y.
Step 2: Since the inverse sine of x is the angle for which the sine equals x, we can change arcsin
(0.9877) = y into sin y = 0.9877.
Step 3: Now we need to look at a table of trigonometric functions that includes degree measures (see
example below) to determine which angle has a sine of 0.9877. The angle whose sine equals 0.9877 is
81º .
(To read this
table, it is important to notice that the table reads from the top as well as the
bottom. The degree values on the left side of the table correspond with the sin,
csc, tan, cot, sec, and cos that read from left to right across the top of the
table. So, the cos of 8º is 0.9903. The degree values on the right side of the
table correspond with the sin, csc, tan, cot, sec, and cos that read from right to
left across the bottom of the table. So, the sin of 81º is 0.9877.)
Step 4: Since 81º has a sine of 0.9877, arcsin (0.9877) = 81º .
Example 5:
Step 1: Set the inverse tangent equal to y.
Step 2: Since the inverse tangent of x is the angle for which the tangent equals x, we can change
Step 3: Consult a table of special trigonometric values (see example
below) to determine which angle has a tangent of
SD 9-12 Miscellaneous Math Topics – Test, Answer Key, Study Guide
Step 4: arctan
Page 17 of 36
Example 6: Find arccos (3.345).
(1) arccos (3.345) = y
(2) cos y = 3.345
(3) arccos (3.345) does not exist
Step 1: Set the inverse cosine equal to y.
Step 2: Since the inverse cosine of x is the angle for which the cosine equals x, we can change
arccos (3.345) = y into cos y = 3.345.
Step 3: The cosine of a number or angle must be in the range from -1 to 1. Since 3.345 is greater than 1,
a number does not exist whose cosine is 3.345; therefore, arccos (3.345) does not exist.
Law of Sines
The law of sines gives the relationship between the lengths of the sides of any triangle and the three
angles of the triangle. The law of sines states that the ratios of the sines of the angles of a triangle to the
lengths of the sides of the triangle are proportional. The law of sines can be used to determine either the
length of a missing side or the degree of an unknown angle.
It is helpful to remember, that side a is always opposite
C.
Example 7: In triangle ABC (above) suppose that a = 20,
for A, C, and c.
SD 9-12 Miscellaneous Math Topics – Test, Answer Key, Study Guide
A, side b is opposite
B, and side c is opposite
B = 42º , and b = 13.40. Solve the triangle
Page 18 of 36
Solving for angle A:
Step 1: Using the law of sines, we can substitute in the values we were given. This procedure would
result in:
We do not need the ratio sin C/c because it does not provide any usable information and it is only
necessary to have two ratios to solve a proportion.
Step 2: Multiply by 20 on each side of the equal sign to get sin A isolated on one side of the equal sign.
Step 3: Determine that the sine of 42º is (0.6691). This can be accomplished using a calculator (first
enter 42, then press the "sin" key OR some calculators use the opposite key sequence) or using a table of
trigonometric functions. Multiply the sine of 42º (0.6691) by 20. Divide that product by 13.40. Sin A =
0.9987.
Step 4: Determine the angle whose sine is 0.9987. This can be accomplished using a table of
trigonometric functions or using a calculator. First enter 0.9987, then press the "arcsin" or
A
= 87º because the sine of 87º is 0.9987.
Solving for angle C:
(1) 180º - (87º + 42º )
(2) C = 51º
Step 1: The measures of the three angles of a triangle always add up to 180º . This knowledge can be
used to determine the measure of C. Subtract the sum of the two known angles (87º and 42º ) from
180º to find the value of the third angle of the triangle.
Step 2: 180º - 129º = 51º , so the measure of angle C is 51º .
Solving for side c:
Step 1: It is only necessary to use two of the three ratios acquired from the law of sines to solve for the
value of c.
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Step 2: Multiply each side of the equal sign by sin 51º to isolate the variable c on one side of the equal
sign.
Step 3: Determine the sine of 51º (0.7771). Multiply the sine of 51º (0.7771) by 20. Divide that product
by the sine of 87º (0.9987). The length of side c is 15.56.
Comparing graphs of trigonometric functions
The standard form for the sine function is y = a sin (kx + b). For purposes of comparing graphs of
trigonometric functions, we will add in a constant c such that y = a sin (kx + b) + c becomes the standard
form. The value of c determines whether the graph of the function shifts upward or downward. The
value of b determines whether the graph shifts to the right or the left. The value of a determines whether
the graph stretches or flattens. If a is negative, the graph is reflected over the x-axis. See the table below.
The table above will also work for cosine functions written in the form y = a cos (kx + b) + c.
Example 8:
(1)
(2) By looking at the table we see that the graph of the function should:
move upward 2 units
should shift to the right
the 1 is positive.
(3) Compare the actual graphs below.
SD 9-12 Miscellaneous Math Topics – Test, Answer Key, Study Guide
the a = 1 does nothing to the graph because
Page 20 of 36
Compound Interest
Compound interest is the interest paid on the initial deposit as well as any interest previously earned. Interest is
usually compounded (paid or figured) more than one time per year.
Biannually (Semiannually) = compounded twice a year
Quarterly = compounded four times a year
Monthly = compounded twelve times a year
Daily = compounded 365 times a year
There is a special formula for determining compound interest.
The initial deposit is called the principal and is denoted with the variable P. The interest rate, denoted
with the variable r, is the percent of the principal that will be earned over a particular period of time,
usually one year. The number of times interest is compounded per year is denoted by the variable n. The
variable t states the number of years (time) the principal is invested. The variable S in the formula
represents the amount of money the account will hold after the interest is accrued over the stated number
of years.
Example 1: If $5,000.00 is invested in an account paying an annual interest rate of 8%, find the amount
of money in the account after 5 years if the interest is compounded quarterly.
Step 1: The principal (initial deposit) is $5,000, so P = $5000. The interest rate is 8%. The interest rate
must be converted from a percent into a decimal, so r = 0.08. The interest is compounded quarterly (4
times per year), so n = 4. The money will be in the account for 5 years, so t = 5.
Step 2: Substitute the known values in place of the variables that represent them.
Step 3: Divide 0.08 by 4, then add that result to 1. Next, 4 and 5 must be multiplied to come up with the
exponent of 20.
Step 4: Applying the order of operations, take 1.02 to the 20th power. This can be accomplished using a
calculator. First, enter 1.02. Then, press the
The next step is to enter the
number 20, and the final step is to press the equal sign. 1.02 to the 20th power equals 1.485947396. Do
not round this number! To finish this problem, multiply 1.485947396 by 5000.
Answer: The amount of money in the account after 5 years is $7,429.74.
Example 2: Genevieve bought an airline ticket for $315.00 with a credit card that charges an annual rate
of 17.9% and compounds the interest monthly. If she does not pay for the ticket for seven months, how
much will she have paid for the ticket?
SD 9-12 Miscellaneous Math Topics – Test, Answer Key, Study Guide
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Step 1: The principal (initial amount invested) is $315.00, so P = $315. The interest rate is 17.9%, so r =
0.179. The interest is compounded monthly and there are 12 months in one year, so n = 12. The airline
ticket is not going to be paid off for 7 months, so the amount of time is 7/12 of a year (t = 7/12).
Step 2: Substitute the known values in place of the variables that represent them.
Step 3: Divide 0.179 by 12, then add the result to the 1. (Note: Do not round until all calculations are
complete.) Next, the 12 and the 7/12 must be multiplied to come up with the exponent of 7.
Step 4: Calculate 1.014916666 to the 7th power, which equals 1.109207228. (Do not round yet.) To
finish this problem, multiply 1.109207228 by 315.
Answer: $349.40
Example 3: How much principal must the Jacob family invest now at 3% compounded monthly in order
to have enough money to buy a $25,000.00 SUV in 5 years?
Step 1: This problem asks the student to solve for P instead of S, so the student should pay close
attention when substituting the values into the formula. The ending amount after the interest is accrued
is $25,000.00, so S = 25,000. The interest rate is 3%, so r = 0.03. The interest is compounded monthly
and there are 12 months in one year, so n = 12. The money will be invested for 5 years, so t = 5.
Step 2: Substitute the known values into the formula.
Step 3: Divide 0.03 by 12 and add 1 to the result. Next, 12 and 5 must be multiplied to come up with
the exponent of 60.
Step 4: Raise 1.0025 to the 60th power using a calculator. The result is 1.161616782.
Step 5: Since this problem needs to be solved for P, divide both sides of the equation by 1.161616782 to
isolate the variable, P.
Step 6: The result is 21,521.72764 which is approximately equal to $21,521.73.
Answer: $21,521.73
Continuously Compounded Interest:
Interest can also be compounded continuously. There is a special formula to follow when compounding
interest continuously.
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The variable A in the formula is the amount in the account after a certain number of years. The variable
P is the principal, r is the annual interest rate, and t is the number of years. The variable e represents the
constant irrational number 2.718281828459... There is a special key on scientific calculators for
calculating e taken to a power. That key is:
Example 4:
(1) Enter 0.05.
(2) Press
OR
(1) Press
(2) Enter 0.05
(3) Press = or ENTER
Answer: approximately 1.051271
Example 5: If Michael invests $900.00 at an annual interest rate of 6% compounded continuously, what
is the amount in the account after 10 years?
Step 1: The principal, P, is 900 because that was Michael's initial deposit. The interest rate is 6%, so r =
0.06, and the money will be in the account for 10 years, so t = 10.
Step 2: Substitute the known values in place of the variables that represent them.
Step 3: Multiply 0.06 by 10 to determine the exponent for e.
Step 4:
calculations are complete.
Answer: $1,639.91
Remember, do not round until all
An activity to help reinforce this concept is to have the student calculate how much money he or she
would have in a savings account using various interest rates, principals, and times. To make the activity
more realistic, students could use the Internet to research available interest rates at various local banks.
Evaluating Solutions - A
The Evaluating Solutions skill asks students to read problems and determine which solutions are most
reasonable.
Evaluating solutions requires a combination of what we know to be true, calculations, estimation, and the
process of elimination.
Example 1: Which description best describes a right triangle?
A.
B.
C.
D.
a figure with 3 sides
a figure with 3 angles
a triangle with a 90º angle
a figure with 3 unequal angles
Solution: We know that a right triangle has 3 sides, 3 angles, and a 90º angle, so either A, B, or C. could be the
possible answer. However, both answer A and B could describe other types of triangles.
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Answer: C because a triangle with a 90º angle MUST be a right triangle.
Example 2: Which word best describes the following numbers:
1/4, 2/5, -3/4, 5/8, 7/9, -1/2
A.
B.
C.
D.
negative fractions
decimals
fractions
whole numbers
Solution: Using the process of elimination, we know that these numbers are NOT decimals or whole numbers.
We also know that these numbers are not all negative fractions.
Answer: C
We may also use calculations and estimation to determine a correct answer choice.
Example 3: Shaqueal wrote an 8-page paper for his English class. The first 4 pages took him 2 hours each to
write. The last 4 pages took him 2 hours total. His paper for social studies took him 8.25 hours to write. What
is the average amount of time he spent working on each of his papers?
A.
B.
C.
D.
8 hours
240 minutes
18 hours
9.125 hours
Solution: We can calculate that his English paper took him 10 hours total (4 x 2 + 2 = 10), and we know his
social studies paper took 8.25 hours. So the average time he spent working on each of his papers is between
8.25 hours and 10 Hours. We can estimate that it is approximately slightly more than 9 hours. From our
calculations and estimations, we eliminate answers A, B, and C.
Answer: D
It may also be helpful to review the formula for the area of a rectangle with the student.
Area = length x width
Trigonometric Identities
A trigonometric identity is a statement that is true for all angle measures. Angle measures can be
represented by variables or by the Greek letter theta An example of an identity is
Note that it is not an equation to be solved. Any value substituted in for theta will
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make the statement true. For example:
There are many different types of trigonometric identities which are often summarized in the
trigonometry section of a textbook. Some of the general categories are listed below as well as a specific
identity for that category:
Note that to correctly and more easily solve trigonometric identity problems requires memorization of
the basic identities.
Example 1: Complete the following identity.
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There is a reciprocal identity that states that
Example 2: Simplify the following expression.
Step 1: Use the Negative identities that state that sin (- x) = - sin x and tan (-x) = - tan x. Substitute the
equivalent expressions to rewrite the original fraction and obtain the fraction on the right.
Step 2: It is often helpful to rewrite all of the expressions in terms of sin x and cos x to make it easier to
determine what the next step should be. In this problem, the reciprocal identity, tan x = sin x/cos x, was
used to rewrite the tan x term in Step 1 and create the new complex fraction of Step 2.
Step 3: The complex fraction of Step 2 can be rewritten to use the conventional division sign, ÷ , to help
determine the next step.
Step 4: A division problem containing fractions can be rewritten as a multiplication problem by
multiplying the first term of the division problem by the reciprocal of the second term. The - sin x is
rewritten as a fraction by making the denominator one (-sin x/1).
Step 5: The sin x terms in the numerator and denominator of the fraction divide out. The product of two
negative quantities is positive, so the product shown in Step 4 simplifies to cos x.
The answer is cos x.
Example 3: Simplify the following.
(1) sin x cos x + cos x sin x
(2) sin x cos x + sin x cos x = 2sin x cos x
The final answer is 2sin x cos x.
Determine the trigonometric functions of an angle when given the sides of a right triangle:
To solve right triangle trigonometry problems, the following ratios need to be used. Note that "opp"
represents the side of the triangle opposite the given angle, A, "adj" represents the side of the triangle
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adjacent to the given angle, A, and "hyp" represents the hypotenuse of the right triangle.
Example 4: Find the sine, cosine, tangent, cosecant, secant, and cotangent for angle ABC.
First identify the specific sides of the right triangle with respect to the given angle. In this problem, the
side opposite angle ABC has length 3, so opp = 3. The side adjacent to angle ABC has length 4, so adj =
4 and the hypotenuse has length 5, so hyp = 5. The ratios are given below.
Example 5: Choose the statement that is true.
To solve this problem, you need to evaluate each of the statements to see if the left side of the statement
can be rewritten to equal the right side.
Evaluate answer choice A:
Step A1: Choice A is false because the Pythagorean Identity states
Evaluate answer choice B:
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Step B1: Replace tan x with sin x/cos x, a reciprocal identity.
Step B2: The cos x terms divide out.
Since sin x does not equal sec x, choice B is false.
Evaluate answer choice C:
Step C1:
Step C2:
Step C3:
C4: One sin x on the bottom of the fraction cancels the sin x on the top of the fraction, leaving
1/sin x.
Step C5: Replace (1/sin x) with (csc x), a reciprocal identity.
Step
Since csc x does not equal sin x, choice C is false.
Evaluate answer choice D:
Step D1: Replace csc x with (1/sin x), a reciprocal identity.
Step D2: The sin x terms in the numerator of Step D1 divide out to get 1.
Step D3: Replace (1/cos x) with sec x, a reciprocal identity.
Since sec x equals sec x, choice D is true.
The correct answer is choice D.
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Credit
Credit is an agreement between a buyer and a seller which states that the buyer will receive the goods before
full payment occurs. Generally there is a fee associated with buying an item with a credit plan, so it is usually
more expensive to buy an item on credit than it is to buy an item with cash. Credit cards also charge a fee. This
fee is called interest and it is in the form of a percentage rate.
Which is Better, Credit or Cash Payment?:
Example 1: A new kayak costs $625. You can pay with cash, or a credit plan with a $125 down
payment and then 24 monthly installments of $25.00. How much more will you pay for the kayak if you
choose the credit plan?
(1) $25.00 x 24 = $600.00
(2) $600.00 + $125.00 = $725.00
(3) $725.00 - $625.00 = $100.00
Step 1: Determine the amount of money you will pay if you choose the credit plan. First multiply $25.00
by 24 to see how much money you will spend on the monthly installments. You will pay $600.00 in
monthly installments.
Step 2: Add the $600.00 to the deposit ($125.00) to determine the amount you will pay for the kayak
using the credit plan. With the credit plan, you will pay $725.00 for the kayak.
Step 3: To determine how much more you will pay with the credit plan, subtract the cash price of the
kayak ($625.00) from the price you will pay using the credit plan ($725.00).
Answer: If you choose the credit plan, you will pay $100.00 more for the kayak than you would if you
paid cash.
Credit Card Interest:
Every credit card charges simple interest. Simple interest is interest charged to an account on a yearly
basis.
Example 2: A credit card account has a balance of $1,234.00 and an annual interest rate of 17.25%.
What is the simple interest that will be charged to the account in 1 year?
(1) $1,234.00 x (17.25%)
(2) $1,234.00x (0.1725) = 212.865
(3) $212.87
Step 1: Multiply the balance of the account by the interest rate.
Step 2: Before multiplying by the interest rate, convert the percent into a decimal number. This involves
moving the decimal point of the interest rate two places to the left.
Step 3: Round 212.865 to the nearest cent. This involves looking at the number in the hundredths place
in 212.865. The 5 to the right of the number in the hundredths place tells us to round the 6 to 7. 212.865
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rounds to $212.87.
Answer: You will pay $212.87 in interest in 1 year.
Example 3: The Gonzales' credit card currently has a balance of $5,256.78. The annual interest rate on
their credit card is 16.9%. They received a credit card offer in the mail for a card with a 6.9% annual
interest rate. How much money will the Gonzales' save in one year if they transfer the balance of their
credit card to the credit card with the lower interest rate?
(1) $5,256.78 x (0.169) = 888.39582 = $888.40 (interest old credit card)
(2) $5,256.78 x (0.069 = 362.71782 = $362.72 (interest new credit card)
(3) $888.40 - $362.72 = $525.68
Step 1: Determine the interest the Gonzales' would have paid with their current credit card. Multiply the
current balance ($5,256.78) by the interest rate (16.9%). Remember to convert the percent into a decimal
before multiplying. 16.9% = 0.169
Step 2: Determine the interest the Gonzales' will pay with the new credit card. Multiply the current
balance ($5,256.78) by the new interest rate (6.9%). Remember to convert the percent into a decimal
before multiplying. 6.9% = 0.069
Step 3: To determine the amount of money the Gonzales' will save, subtract the interest they would pay
with the new credit card ($362.72) from the interest they would pay with the old credit card ($888.40).
Answer: The Gonzales' will save $525.68 in one year if they transfer their credit card balance to the new
credit card.
Dividing Integers
Integers are the set of positive and negative whole numbers, including zero. To find the quotient (answer to a
division problem) of two integers, the following rules apply:
The quotient of two integers with different signs is negative. Example: 16 ÷ -4 = -4.
The quotient of two integers with the same sign is positive. Examples: 16 ÷ 4 = 4 and -16 ÷ -4 = 4.
Operations within parentheses are completed first. After performing operations within parentheses, perform all
multiplication and division in order from left to right. The last step is to perform all addition and subtraction in
order from left to right. (It may be helpful here to review order of operations and/or multiplying with integers.)
Example 1: 4(-3 x 2) ÷ (12 ÷ 2) = ?
(1) -3 x 2 = -6 and 12 ÷ 2 = 6
(2) 4(-6) ÷ 6 = ?
(3) -24 ÷ 6 = ?
(4) -24 ÷ 6 = -4
Step 1: Perform operations within parentheses: ( -3 x 2 = -6) and (12 ÷ 2 = 6).
Step 2: Write out the problem, replacing the values within the parentheses with the new values.
Step 3: Perform multiplication or division in order from left to right. Multiply first because it comes first when
reading from left to right. 4(-6) = -24.
Step 4: Divide -24 by 6 to get -4. Remember the quotient of two integers with different signs is negative.
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Answer: -4
The following example illustrates the use of rules for dividing integers using "is greater than" (>) and "is less
than" (<).
Example 2: -24 ? 3(10 ÷ -2)
(1) 10 ÷ -2 = -5
(2) -24 ? 3(-5)
(3) 3 x -5 = -15
(4) -24 ? -15
(5) -24 < -15
Step 1: Perform operations within parentheses. 10 ÷ -2 = -5.
Step 2: Rewrite the problem with -5 in place of the parentheses.
Step 3: Multiply 3 x -5 to get -15.
Step 4: Rewrite the problem with -15 in place of 3(-5).
Step 5: To determine which symbol to place between -24 and -15, think of the integers as being money. -24
would be like owing someone $24 and -15 would be like owing someone $15. Since owing $24 is more in debt
than owing $15, -24 is less than -15.
Answer: -24 < -15
Integers: Multiple-step Computation
Integers are positive and negative whole numbers, including zero.
Before computing with integers, let's first review the rules of operations on integers.
When adding two integers with the same sign, add their absolute values. Then give the sum (answer) the sign
of the integers.
-3 + -2 = ?
|-3| + |-2| = ?
3 +2 = 5, then make the result negative.
Answer: -5
When adding integers with different signs, first find their absolute values. Then subtract the lesser absolute
value from the greater absolute value, and give the result the sign of the integer with the greater absolute value.
-7 + 3 = ?
|-7| = 7 and |3| = 3 (find the absolute values)
7 - 3 = ? (subtract the lesser from the greater)
7-3=4
-7 + 3 = -4 (The result is given the sign of the greater integer.)
Subtracting integers is the same as adding the opposite.
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3 - -7 = ?
3 + +7 = 10 (add the opposite)
3 + 7 = 10
When multiplying integers, the product (answer) of two integers with the same sign is positive. The product of
two integers with different signs is negative.
(-5)(-6) = 30
(-5)(6) = -30
When dividing integers, the quotient (answer) of two integers with the same sign is positive. The quotient of
two integers with different signs is negative.
-6 ÷ -3 = 2
-6 ÷ 3 = -2
When performing computations with more than one operation, follow the rules for the order of operations.The
order of operations is as follows:
1. Perform operations within parentheses, braces, or brackets.
2. Multiply and divide from left to right
3. Add and subtract from left to right
Example 1: (-2 - 5) + 10 = ?
(1) -2 - 5 = -7
(2) -7 + 10 = ?
(3) -7 + 10 = 3
Step 1: Perform the operation within parentheses, -2 - 5 = -7.
Step 2: Replace the -7 for the value of the parentheses.
Step 3: Complete the addition problem.
Answer: 3
When calculating problems with <, >, and =, perform the calculations on each side to determine the values of
each side.
Example 2: -(57 - -10) ? (-12 - -22)
(1) -(57 - -10) = -(57 + 10) = -(67) = -67
(2) -12 - -22 = -12 + 22 = 10
(3) -67 ? 10
(4) -67 < 10
Step 1: Evaluate the value in the first set of parentheses. Remember, subtracting a negative is the same as
adding the opposite. Then distribute the negative sign to the answer.
Step 2: Evaluate the value in the second set of parentheses.
Step 3: Rewrite the mathematical sentence with the new values.
Step 4: Negative 67 is less than positive 10.
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Answer: <
Trigonometric Ratios - B
Trigonometry is the study of triangle measurement. A trigonometric ratio involves the ratio of the lengths of
the sides of a right triangle.
Trigonometric ratios allow you to find the lengths of the sides of right triangles using one of their acute
angles. Acute angles are angles which measure less than 90 degrees. A right triangle has one angle
which measures 90 degrees. The triangle side opposite the 90 degree angle is called the hypotenuse.
The hypotenuse is the longest side of a right triangle.
Utilize this information to draw a right triangle. Label the 90 degree angle, Angle C. Label the opposite
side, the hypotenuse, Side C. Label the acute angle above or below Angle C, Angle B. Label the side
across from Angle B, Side B. Label the remaining angle, Angle A. Label the side opposite Angle A,
Side A. Now assign measures: Side C = 5, Side B = 3, Side A = 4.
Your triangle should look like the triangle below.
The formulas for trigonometric ratios are:
Sine of an acute angle = measure of the opposite leg/measure of the hypotenuse
Cosine of an acute angle = measure of the adjacent leg/measure of the hypotenuse
Tangent of an acute angle = measure of the opposite leg/measure of the adjacent leg
Therefore, the trigonometric ratios for angle A of the triangle you've drawn are:
sin A = 4/5
cos A = 3/5
tan A = 4/3
Students should be familiar with the Pythagorean Theorem which often needs to be applied in order to
complete the third side of a right triangle prior to finding trigonometric ratios. The theorem asserts that
in a right triangle the square of the hypotenuse is equal to the sum of the squares of the legs.
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Example 1:Triangle ABC is a right triangle. Find the cosine of angle A. Rationalize the denominator.
Step 1: Determine the length of side y. Apply the Pythagorean Theorem.
Step 2: Substitute the values into the cosine formula.
Step 3: Rationalize the denominator.
Answer:
Solving Right Triangles - B
Solving right triangles involves using the trigonometric ratios of sine, cosine, and tangent. Students should be
able to use these ratios and trigonometric tables to find the lengths of unknown sides of right triangles and
measures of missing angles to the nearest degree.
Trigonometric ratios allow you to find the lengths of the sides of right triangles using one of their acute
angles. Acute angles are angles which measure less than 90 degrees. A right triangle has one angle
which measures 90 degrees. The triangle side opposite the 90 degree angle is called the hypotenuse.
The hypotenuse is the longest side of a right triangle.
Utilize this information to draw a right triangle. Label the 90º angle, C. Label the opposite side, the
hypotenuse, side c. Label the acute angle above or below C, B. Label the side across from B, side
b. Label the angle to the side of C, A. Label the side opposite A, side a. Now assign measures:
side c = 5, side b = 3, side a = 4.
Your triangle should look like the triangle below.
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The formulas for trigonometric ratios are:
Sine of an acute angle = measure of the opposite leg/measure of the hypotenuse
Cosine of an acute angle = measure of the adjacent leg/measure of the hypotenuse
Tangent of an acute angle = measure of the opposite leg/measure of the adjacent leg
Therefore, the trigonometric ratios for angle A of the triangle you've drawn are:
sin A = 4/5
cos A = 3/5
tan A = 4/3
Apply this knowledge to the following problem.
Example 1: In right triangle ABC,
Side A?
A measures 28º and Side B measures 21cm. What is the length of
(1) tan A = Side A/Side B
(2) tan 28º = Side A/21
(3) 0.53 = Side A/21
(4) (0.53 x 21) = Side A
(5) Side A = 11.13
Step 1: Find the trigonometric ratio formula that uses the side and angle information (tangent). Apply
the known information to the formula.
Step 2: Look up the tangent of a 28 degree angle on a table of trigonometric ratios. Find the tangent to
the nearest tenth degree. (See below for directions to using a table.) The tangent of a 28 degree angle is
0.53. Apply this to the equation.
Step 3: Determine the tangent of 28º and rewrite the equation with that value in place of tan 28º .
Step 4: Cross multiply.
Step 5: Side A of the triangle is 11.13cm.
To read a trigonometric table you need to know either the measure of an acute angle, or the sine, cosine,
or tangent of the angle. The following table shows a portion of the Table of Trigonometric Ratios.
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The sine (sin) of a 12º angle is 0.2079. The angle degree that has a tangent (tan) of 0.2493 is 14.
There are many real world applications for triangles and using trigonometric ratios and tables to solve
right triangles. The following is an example.
Example 2: A flag pole casts a 12 foot shadow at 3:00 p.m., when the angle of elevation of the sun
measures 56º . Find the height of the flag pole (side a) to the nearest hundredth.
Draw a flag pole and the sun above. Then, draw a short line from the bottom of the flag pole and make
a point (a period). Mark this line as the 12 foot shadow. (Make sure that the sun is on the opposite side
of the flag pole as the shadow.) Draw a dotted line from the top of the flag pole to the point to complete
the right triangle. Label the side which indicates the flag pole, Side A.
The 90 degree angle is the angle created by the bottom of the flag pole and the line which indicates the
flag pole's shadow. The dotted line is the hypotenuse. We know that the angle of elevation of the sun to
the flag pole is 56 degrees. This means that the angle created by the point which indicates the 12 foot
shadow and the hypotenuse is a 56 degree angle.
(1) tan 56º = Side a/12
(2) 1.48 = Side a/12
(3) (1.48 x 12) = Side a
(4) Side a = 17.76
Step 1: Substitute the known information into the tangent formula.
Step 2: Look up the tangent of 56º in a table of trigonometric ratios and find the measure to the nearest
hundredth.
Step 3: Cross-multiply.
Step 4: Determine the length of Side a.
The height of the flag pole to the nearest hundredth is 17.76 feet.
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