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Chapter 10
Contents
10.1 Right Triangle Trigonometry
10.2 Evaluating Trigonometric
Functions
10.3 Inverse Trigonometric
Functions
10.4 Graphing Trigonometric
Functions
10.5 The Law of Sines and Law of
Cosines
10.6 Verifying Trigonometric
Identities
10.7 Angle Sum and Difference
Identities
10.8 Double-Angle and HalfAngle Identities
10.9 Solving Trigonometric
Equations
Math Labs
The Circle of Your City
The Sun Today
The Sine Curve of Biorhythms
Swing of a Pendulum
Math Applications
Chapter Review
Chapter Assessment
434 Chapter 10 Trigonometric Functions and Identities
Trigonometric Functions
and Identities
Why should I learn this?
Trigonometry is about right angles and angle relationships. Many real‑world
problems that are solved using trigonometric functions relate to indirect
measurement. The topics of this chapter incorporate previously learned skills,
such as using the Pythagorean Theorem and simplifying radicals.
Trigonometric functions and relationships are the underlying mathematics used
in occupations that deal with distance and measurement.
• Air Traffic Controllers interpret data processed by machines that are
programmed using trigonometric functions.
• Cartographers use trigonometry to indirectly measure distances and
verify that their maps are accurate.
• Musicians, especially those who play stringed instruments, produce
harmonic tones that are based on variations of the sine function.
In this chapter, you will use the trigonometric ratios to interpret and solve
problems. You will also learn to convert measurements between degrees
and radians.
Project Idea: Shadow to Shadow
When you are outside on a sunny day, you cast a shadow. The length
of the shadow depends on the time of day. The ratio of the height of
a person to the length of the shadow cast is equivalent to the ratio of
another person or object and its shadow length. Both measurements
must be taken at the same time of day from the same place.
Select an object, such as a tree or a flag pole, on the school
grounds. Work with a partner to determine and verify the
height of that object using indirect measurement. Find the
length of shadows cast by you and your selected object.
Then use your height and measurements gathered to find
the height of your object. Your partner should make the
same measurements and computations using his or her
height and shadow to find the height of the object. You
can verify the height of your object by comparing results
with your partner.
Chapter 10 Trigonometric Functions and Identities
435
Lesson 10.1 Right Triangle Trigonometry
Objectives
Find lengths in triangles
using trigonometric
relationships.
Activity
From where Horacio stands at a distance of 38 feet from the base of
the school building, the angle to the top of the school is 40°. What is
the height of the school building?
Finding Right Triangle Ratios
Use a protractor and a straightedge to
draw three right triangles on a sheet
of paper. Each triangle should have an
acute angle that measures 50°. Label
this angle a in each triangle. Make
the hypotenuses of the three similar
triangles 5, 8, and 10 centimeters.
1 Measure the lengths of the legs of each right triangle to the
nearest tenth. Small: 3.8 cm, 3.2 cm; Medium: 6.1 cm, 5.1 cm;
Large: 7.7 cm, 6.4 cm
2 What is the ratio of the length of the leg opposite ∠a to
the length of the hypotenuse in each triangle? Round to the
nearest tenth. 0.8 for each triangle
3 What is the ratio of the length of the leg adjacent to ∠a to
the length of the hypotenuse in each triangle? Round to the
nearest tenth. 0.6 for each triangle
4 What is the ratio of the length of the leg opposite ∠a to the
length of the leg adjacent to ∠a in each triangle? Round to
the nearest tenth. 1.2 for each triangle
5 Make a conjecture about these ratios for all right triangles
that have an acute angle of 50°. The ratios are the same for any
right triangle with an acute angle of 50°.
436 Chapter 10 Trigonometric Functions and Identities
Trigonometric Ratios
There are six special ratios between the lengths of the sides of right triangles
that are known as trigonometric ratios. These trigonometric ratios are
shown below.
Trigonometric Ratios
The sine of angle A, abbreviated sin A =
leg opposite ∠A a
=c
hypotenuse
The cosine of angle A, abbreviated cos A =
leg adjacent ∠A b
= c
hypotenuse
leg opposite ∠A a
=
leg adjacent ∠A b
1 = hypotenuse = c
The cosecant of angle A, abbreviated csc A =
sin A leg opposite ∠A a
The tangent of angle A, abbreviated tan A =
1 = hypotenuse = c
cos A leg adjacent ∠A b
1 = leg adjacent ∠A = b
The cotangent of angle A, abbreviated cot A =
tan A leg opposite ∠A a
The secant of angle A, abbreviated sec A =
Notice that the sine and cosecant ratios are multiplicative inverses of each
other, as are the cosine and secant ratios and tangent and cotangent ratios.
Example 1 Finding Trigonometric Ratios
Find the trigonometric ratios for ∠N in the triangle below.
10.1 Right Triangle Trigonometry
437
Solution
Use the definitions of the trigonometric ratios. Reduce each ratio.
sin N = 8 = 4
10 5
csc N = 10 = 5
8 4
cos N = 6 = 3
10 5
sec N = 10 = 5
6 3
tan N = 8 = 4
6 3
cot N = 6 = 3
8 4
Ongoing Assessment
Find the trigonometric ratios
for ∠Z in the triangle below.
see margin
Example 2 Finding the Height of a Building
Use the information in the opening paragraph of this lesson. What is the
height of the school building?
Solution
Draw a picture to help solve the problem.
Choose a trigonometric ratio that relates the leg opposite an angle and the
leg adjacent to an angle. Use the tangent ratio to find the unknown height of
the school building.
x
tan 40° = 38
Use a calculator to approximate tan 40° and solve for x.
0 .839≈ x
38
31.9 ≈x
The height of the school building is about 31.9 feet.
438 Chapter 10 Trigonometric Functions and Identities
Ongoing Assessment
The angle from the top of a tree to a point 20 meters from the base of
the tree is 52°. What is the height of the tree? Round your answer to the
nearest tenth. about 15.6 m
Lesson Assessment
Think and Discuss
see margin
1. Which trigonometric ratio in a right triangle relates an angle
measure to the side opposite the angle and the hypotenuse?
2. Explain how to find the height of a tree if the horizontal distance
to the base of the tree and the angle of elevation to the top of
the tree are known.
3. In a right triangle, what ratio of side lengths is equal to the cosine
of an angle?
4. What happens to the cosine of an angle if the side lengths of the
triangle are doubled?
5. Which trigonometric ratio is the reciprocal of the sine ratio?
Practice and Problem Solving
Use triangle DEF to find each trigonometric ratio.
6.sin E
5
13
9.cos E
12.sec D
15.csc D
12
13
13
5
13
12
7.cos D
10.sin D
13.csc E
16.cot E
5
13
12
13
13
5
12
5
8.tan E
11.tan D
14.cot D
17.sec E
5
12
12
5
5
12
13
12
10.1 Right Triangle Trigonometry
439
Solve for x in each right triangle. Round your answer to the
nearest tenth if necessary.
18.
19.
x = 6.0
20.
x = 8.2
21.
x = 8.5
x = 30.1
22. The angle from the top of a flagpole to a point 30 feet from the
base of the flagpole is 54°. What is the height of the flagpole?
Round your answer to the nearest tenth. 21.8 ft
23. Three fishing docks are situated on a riverbank as shown. What is
the width of the river? Round your answer to the nearest tenth.
54.5 yd
24. A ladder is leaning against the side of the house. The ladder and
the side of the house form an angle of 25°. The bottom of the
ladder is 5.1 feet from the base of the house. What is the length
of the ladder? Round your answer to the nearest tenth. 12.0 ft
25. After an airplane takes off from the runway, it climbs at an
angle of elevation of 25°. What is the altitude of the plane after
covering a horizontal distance of 400 feet? Round your answer to
the nearest tenth. 186.5 ft
440 Chapter 10 Trigonometric Functions and Identities
26.
What is the height of the skyscraper to the nearest meter? 192 m
27. Carlos built a bike ramp shaped like a right triangular prism. The
board used for the ramp is 6 feet long. If the ramp has an angle
of elevation of 24º, what is the height of the ramp? Round your
answer to the nearest tenth. 2.4 ft
28. A wire support for a radio antenna forms a 68º angle with the
side of the antenna tower. The wire is 48 feet long. How high
up the antenna tower does it reach? Round your answer to the
nearest tenth. 18.0 ft
29. What is the horizontal distance between the hot air balloon and
the landing zone? Round your answer to the nearest tenth.
2,425.8 ft
10.1 Right Triangle Trigonometry
441
30. A wheelchair ramp is 20 feet long with an angle of incline of 5°.
How many feet does the incline of the ramp rise? Round your
answer to the nearest tenth. 1.7 ft
31. The angle of elevation from a boat to the top of a lighthouse
is 21°. The height of the lighthouse is 47 feet. What is the
distance from the boat to the base of the lighthouse? Round to
the nearest tenth of a foot. 122.4 ft
32. A landscaper is standing 25 meters from the base of a tree. The
angle of elevation from the landscaper to the top of the tree
is 41°. What is the height of the tree? Round to the nearest tenth
of a meter. 22.7 m
Mixed Review
Find each difference.
33. 5 – 2 32x − 6 x x + 3x
x +3
27 − 2k
35. 9 – 2k
k + 2 3k + 6 3k + 6
34. 4 – 1 3240−r5r
5r
8
2
36. t – t − 7 3t +2 3t + 28
4t + 16t
t+4
4t
37. Find the length of a side of a rectangle if its area can be
modeled by the expression 28x2 + 10x – 2 and its width by
the expression 4x + 2. 7x – 1
38. If the third term in the expansion of (x + y)n has a coefficient
of 15, what is the value of n? n = 6
442 Chapter 10 Trigonometric Functions and Identities
Lesson 10.2 Evaluating Trigonometric
Functions
Objectives
Convert between
degrees and
radians.
Find the length of
an arc.
Evaluate
trigonometric
functions.
Radians and Degrees
To draw an angle in standard
position, place the vertex at the
origin and one ray of the angle
along the positive x-axis. The ray
on the x‑axis is the initial side of
the angle, and the other ray is the
terminal side of the angle.
To measure an angle in standard
position, find the amount of
rotation counterclockwise from
the initial side to the terminal
side. Angles can be measured
in degrees and in radians. A
complete rotation is 360° or
2π radians.
Example 1 Converting Radians and Degrees
Find the radian measure of an angle with a measure of 120°.
Solution
A full rotation of 360° is equal to 2π radians. Set up and solve a
proportion for the angle measure in radians.
120°
= r radians
360° 2π radians
120 • 2π = 360° • r
240π = r
360
2π = r
3
An angle with a measure of 120° has a measure of 2≠ , or
3
approximately 2.1 radians.
442 Chapter 10 Trigonometric Functions and Identities
Arc Length
The unit circle has a radius of 1 unit
and is centered about the origin.
A central angle of a circle is an
angle with a vertex at the center of
the circle. An intercepted arc is the
part of the circle that has endpoints on
the sides of the central angle and all
points in the interior of the angle.
Arc Length
If a circle with radius r has a central angle of measure
θ radians, the length, l, of the intercepted arc is l = rθ.
Example 2 Measuring Arc Length
Find the length of the intercepted arc
in the circle.
Solution
The measure of the central angle is 1.9 radians, and the circle has a radius of
6 centimeters. Use the formula for the length of an intercepted arc.
l = rθ
l = (6)(1.9)
l = 11.4
The length of the intercepted arc is 11.4 centimeters.
10.2 Evaluating Trigonometric Functions
443
Ongoing Assessment
The measure of the central angle is 105°, and the circle has a radius of
≠ in.
5 inches. Find the length of the intercepted arc in the circle. 35
12
Trigonometric Functions
The sine function ( f (θ ) = sin θ )
takes an angle measure as its
input and gives the corresponding
y-coordinate where the terminal
side of the angle intersects
the unit circle. The cosine
function ( f (θ ) = cos θ ) gives the
corresponding x-coordinate where
the terminal side of the angle
intersects the unit circle.
The tangent function
( f (θ ) = tan θ ) is the y-coordinate
of the point where the ray of
the terminal side of the angle θ
intersects the tangent line x = 1.
Example 3 Finding Trigonometric Values
Use the circle to evaluate
f (θ ) = sin θ for θ = 30°.
Solution
The sine function returns the y-coordinate of the point on the unit
circle intersected by the terminal side. So sin 30° = 0.5.
444 Chapter 10 Trigonometric Functions and Identities
Using a Calculator
A scientific calculator can be used to evaluate trigonometric functions of
different angles. Make sure your calculator is in degree mode if the angle
measure is given in degrees and in radian mode if the angle measure is given
in radians.
Example 4 Using a Calculator
Use a calculator to evaluate y = tan θ for θ = 160°. Round your answer to
the nearest thousandth.
Solution
Be sure the calculator is in DEGREE mode. Then press the
key and
enter the angle measure, 160. Note that on some scientific calculators, the
measure of the angle is entered first.
tan 160° ≈ –0.364
Ongoing Assessment
Use a calculator to evaluate each trigonometric expression. Round to the
nearest thousandth. Be sure the calculator is in RADIAN mode. see margin
a. cos 2≠ 7
b. sin ≠ 4
c. tan 2p
Lesson Assessment
Think and Discuss
see margin
1. Explain how to convert an angle measure from radians
to degrees.
2. Describe how to find the measure of a central angle if the
circumference of the circle and the length of the intercepted arc
are known.
3. What is the x-coordinate of the point where the terminal side of
an angle, θ, intersects the unit circle?
4. Describe in your own words what the value of tan θ means as it
pertains to the unit circle.
5. What is the y-coordinate of the point where the terminal side of
an angle, θ, intersects the unit circle?
6. Which trigonometric function has a value of 1 for θ = 45°?
10.2 Evaluating Trigonometric Functions
445
Practice and Problem Solving
Convert each angle measure from degrees to radians.
7.180° π radians
9.60°
≠ radians
3
8.135°
10.75°
3≠ radians
4
5≠ radians
12
Convert each angle measure from radians to degrees.
11. ≠ 90°
12. 3≠ 67.5°
2
8
13. 3≠ 270
14. 7≠ 157.5°
2
8
Use the unit circles to evaluate each trigonometric function.
Round each answer to the nearest thousandth.
15.
cos 60° 0.5
16. sin 3≠
2
–1
17.
tan 40° 0.839
18. sin 5≠
6
0.5
446 Chapter 10 Trigonometric Functions and Identities
Find the length of the intercepted arc in each circle. Round your
answer to the nearest tenth if necessary. see margin
19.
20.
Use a calculator to evaluate each trigonometric function. Round
to the nearest thousandth when necessary.
21. cos 2≠ –0.5
22. sin 85° 0.996
3
23. cos 90° 0
24. tan 3≠ –1
4
25. sin 225° –0.707
26. tan 135° –1
Use a calculator to evaluate each function for the given
argument. Round to the nearest thousandth when necessary.
27. y = 3cos 2θ, θ = 20º
2.298
28. g(θ) = 4 – sin 1.5θ, θ = 5≠ 3.076
12
29. The height above the ground of a person riding on a Ferris Wheel
can be modeled by the function f(x) = 30 – 28cos ≠x . In the
24
function, x is the number of seconds since the ride began, and
f(x) is the height in feet.
a. What is the height of the passenger when the ride begins
(at x = 0 seconds)? 2 ft
b. The passenger reaches the top of the Ferris Wheel after
24 seconds. What is the height at the top of the ride? 58 ft
c. What is the passenger’s height above the ground after
10 seconds? Round to the nearest tenth if necessary. 22.8 ft
d. How long does it take for the passenger to complete one
full rotation around the Ferris Wheel? Check your answer by
evaluating the trigonometric function. 48 s
10.2 Evaluating Trigonometric Functions
447
30. A stretch of highway has a 12° angle of incline. What is this angle
≠ radians
measure in radians? 15
31. Kimmy skated from point A to point B along the outside of the
skating rink. How far did she travel? Round your answer to the
nearest foot. 314 ft
32. The temperature, in degrees Fahrenheit, inside a refrigerator can
be modeled by the function y = 38 + 1.5cos ≠x , where x is the
10
number of minutes after the refrigerator’s motor begins to run.
a. The warmest temperature inside the refrigerator occurs just as
the motor begins to run. What is this temperature? 39.5ºF
b. The motor runs for 10 minutes until the air inside the
refrigerator is cooled to its lowest temperature. Then the
motor rests for 10 minutes before it begins to run. What is
the coolest temperature inside the refrigerator? 36.5ºF
33. Nancy attaches a reflector to one of the spokes of her bicycle tire
for increased visibility in low light. The position of the reflector
relative to the ground as Nancy pedals down the street is given
by the function h(x) = 10 + 7sin 4πx. In the function, x is the
number of seconds since Nancy began pedaling, and h(x) is the
height, in inches, of the reflector above the ground.
a. How long does it take for the reflector to complete one
full rotation? Verify your answer by substituting it into the
function. 0.5 s
b. What is the maximum height of the reflector above the
ground? 17 in.
Mixed Review
State whether or not each infinite geometric series converges
or diverges. Then determine whether or not each series has
a sum.
d
∞
34. ∑ 5 1 converges; has a sum
2
d =1
b
∞
35. ∑ 0.45 10 diverges; does not have a sum
9
b=1
()
( )
36. A radio tower is located 10 miles east and 15 miles north of the
origin on a map of Reedsburg. The tower has enough power to
broadcast over a radius of 40 miles. Write an equation to model
the area of the broadcast of the radio tower.
(x – 10)2 + (y – 15)2 = 402
448 Chapter 10 Trigonometric Functions and Identities
Lesson 10.3 Inverse Trigonometric
Functions
Objectives
Find the angle measure
given the side lengths of
a triangle.
Joshua and his friends built a skateboarding ramp. The ramp is 5 feet
long and 2 feet high. What is the angle of the ramp’s incline?
Evaluating Inverse Trigonometric Functions
An inverse function is a mathematical function that “undoes”
another function. That is, if f and f –1 are inverse functions,
(f ◦ f –1)(x) = x and (f –1 ◦ f )(x) = x. For the sin, cos, and tan functions,
the corresponding inverse functions are sin–1, cos–1, and tan–1.
Example 1 Evaluating Inverse Trigonometric
Functions
Use a calculator to find the measure, in radians, of an angle that has a
sine value of 0.707. Round your answer to the nearest hundredth.
Solution
Be sure the calculator is in RADIAN mode. Enter the sine value on
the calculator. Then use the sin–1 function to evaluate the inverse
trigonometric function.
sin–1 (0.707) ≈ 0.79 radians
Ongoing Assessment
Use a calculator to find the measure, in degrees, of an angle that has
a tangent value of 2.145. Round your answer to the nearest whole
degree. Be sure the calculator is in DEGREE mode. 65°
10.3 Inverse Trigonometric Functions
449
Finding Angle Measures
Inverse trigonometric functions can be used to find missing angle measures
when you know the side lengths of a right triangle.
Example 2 Finding an Angle Measure
Given Side Lengths
Use the information from the opening paragraph of this lesson. What is the
measure, in degrees, of the angle of the ramp’s incline? Round your answer
to the nearest tenth.
Solution
Draw a diagram.
Recall that the tangent of an angle in a right triangle is defined as the ratio of
the length of the side opposite the angle to the length of the side adjacent to
the angle. So, tan x = 2 . Use the inverse tangent function to solve for x.
5
tan x = 2
5
()
tan −1 (tan x) = tan −1 2
5
x ≈ 21.8°
The angle of the ramp’s incline, rounded to the nearest tenth, is 21.8°.
Ongoing Assessment
The figure shows the cross-section of a roof. What is the roof’s angle of
incline, x, in degrees? Round your answer to the nearest tenth. 22.6°
450 Chapter 10 Trigonometric Functions and Identities
see margin
After taking off, a commercial jet covers
a horizontal distance of 23,100 feet
during the first 45 seconds of flight
and climbs to a height of 3,040 feet
above the ground. What is the plane’s
angle of elevation during the first part
of the ascent. Explain why it might be
important to know the plane’s initial
angle of elevation.
Step 1 Understand the Problem
What is the relevant information
presented in the problem statement?
What will you need to calculate to solve
the problem?
Step 2 Develop a Plan
Problem-solving strategy: Use a diagram.
What kind of triangle can be used to model
the situation? Drawing a diagram can help
you understand the problem and develop a
method to solve it.
Step 3 Carry Out the Plan
Model the problem situation with a diagram.
Use the diagram to help you set up and solve
a trigonometric equation for the jet’s angle of
elevation, x. Round your answer to the nearest
tenth of a degree.
Step 4 Check the Results
Check the angle measure by substituting it into
the equation and making sure the result is a
true number sentence.
Lesson Assessment
Think and Discuss
see margin
1. What is the result when you simplify tan–1(tan x)? Explain.
2. Explain how to find the measure of the angle whose terminal side
intersects the unit circle at (0.5, 0.866).
3. If the lengths of the legs of a right triangle are known, how can
the measures of the acute angles of the triangle be found?
4. Which inverse trigonometric function would you use to find the
measure of an acute angle in a right triangle if you know the
lengths of the hypotenuse and the side opposite the angle?
5. How many values of x are there such that tan x = 0? Explain.
10.3 Inverse Trigonometric Functions
451
Practice and Problem Solving
Find the measure, in degrees, of each angle x. Round to the
nearest tenth if necessary.
6.sin x = 0.65, –90º ≤ x ≤ 90º 40.5º
7.cos x = 0.5, 0º ≤ x ≤ 180º 60º
8.sin x = 1, –90º ≤ x ≤ 90º 90º
9.tan x = 2.35, –90º < x < 90º 66.9º
10.cos x = 0.26, 0º ≤ x ≤ 180º 74.9º
Find the measure, in radians, of each angle x. Round to the
nearest thousandth if necessary.
11.cos x = 0.809, 0 ≤ x ≤ π 0.628 radians
12.sin x = 0.707, − π ≤ x ≤ ≠ 0.785 radians
2
2
π
≠
13.tan x = 11.6, − < x <
1.485 radians
2
2
14.sin x = 0, − π ≤ x ≤ ≠ 0 radians
2
2
15.cos x = 0, 0 ≤ x ≤ π 1.571 radians
16.tan x = 1, − π < x < ≠
2
2
0.785 radians
Solve for the missing angle in each right triangle. Round your
answer to the nearest tenth if necessary.
17.
19.
66.8º
452 Chapter 10 Trigonometric Functions and Identities
18.
46.2º
11.3º
20. What is the angle of descent of the airplane? Round your answer
to the nearest tenth if necessary. 22.6º
21. In triangle WXY, m∠W = 90º, x = 6.4, and y = 3.7. Find m∠X,
in radians, to the nearest thousandth. 1.047 radians
22. One of the legs of an isosceles right triangle is 10 units long and
the base is 10 2 units long. What is the measure, in degrees,
of the base angles of the triangle? Round to the nearest tenth
if necessary. 45º
23. Give three angle measures, in degrees, that have a cosine value of
–0.5. Round your answers to the nearest tenth if necessary.
Answers will vary. sample answers: 120º, 240º, 480º
Mixed Review
24. The geometric sequence $500,
, $551.25, $578.81 gives
the amount of money in Fran’s money market at an interest rate
of 5% compounded quarterly. What is the missing amount in
the sequence? $525
25. Japanese magnetic levitation trains (maglev) are among the
fastest trains in the world. A maglev train sounds a 320-Hz
whistle while traveling at its top speed. What is the speed of the
train if the whistle is perceived as 550 Hz? Round to the nearest
tenth. (Assume that the speed of sound is 760 miles per hour.)
≈ 317.8 mph
10.3 Inverse Trigonometric Functions
453
Lesson 10.4 Graphing Trigonometric
Functions
Objectives
Graph trigonometric
functions.
A floating object is observed to
move in circles when waves oscillate
harmoniously in deep water. When
the object is placed in deeper water, it
moves along a circular path. The size
of the circular path gets smaller and
smaller in diameter at different water
depths. At a certain depth, the object
would stand still. This is the wave’s
base, which is exactly half the wave’s
length. An oceanologist is studying the
position of a leaf floating on the surface
of a wave. The oceanologist describes
the motion of the leaf by the function y = 1.5sin 2θ. The oceanologist
wants to graph this function to display the motion of the leaf.
The Sine Function
Recall that the function, y = sin θ, takes an angle measure as its input
and gives the corresponding y-coordinate where the terminal side of
the angle intersects the unit circle. So the domain of the function is
the set of all angle measures, θ, and the range is between –1 and 1.
The graph of a sine function is called a sine curve.
The amplitude of a function is half the positive difference between
its maximum and minimum values. So the amplitude of the function
y = sin θ is 1. The period of a function is the horizontal length of one
cycle. The function y = sin θ completes one full cycle as its output
values range from 0 to 1 to –1 and then back to 0. So the period of the
standard sine curve is 360° or 2π radians.
454 Chapter 10 Trigonometric Functions and Identities
The y-values on the unit circle are positive in Quadrants I and II, and
negative in Quadrants III and IV. Thus, the sine curve is positive between
0° and 180° and is negative between 180° and 360°. Use a calculator to
verify the points along the sine curve.
Characteristics of Sine Functions
For the generalized sine function y = asin bθ, with a ≠ 0, b > 0,
and θ in radians,
• the amplitude of the function is |a|.
• the number of cycles in the interval from 0 to 2π radians is b.
• the period of the function is 2π .
b
Example 1 Graphing Sine Functions
At the beginning of this lesson information was given about an oceanologist
describing the motion of a leaf floating on the surface of a wave. The function
used was y = 1.5sin 2θ. Graph this function over the interval 0 ≤ θ ≤ 2π.
Solution
The amplitude of the function is 1.5 because a = 1.5. Because b = 2, the
function will complete 2 cycles in the interval 0 ≤ θ ≤ 2π. The period, or
horizontal length, of the function is 2≠ = 22≠ = ≠ . Choose appropriate scales
b
for the vertical and horizontal axes. Then graph the function. The graph will
complete 2 cycles and range from –1.5 to 1.5.
Ongoing Assessment
Graph a sine curve that has amplitude 2 and a period of 4π over the interval
0 ≤ θ ≤ 4π.
10.4 Graphing Trigonometric Functions
455
The Cosine Function
The cosine function y = cos θ gives the corresponding x-coordinate where
the terminal side of the angle intersects the unit circle. So the graph of the
cosine function is a sine curve that is shifted to the left. A phase shift is a
horizontal translation of a periodic function. The cosine and sine functions
have graphs that are 90° or ≠ radians out of phase.
2
Characteristics of Cosine Functions
The generalized cosine function y = acos bθ, with a ≠ 0,
b > 0, and θ in radians has the following properties:
• the amplitude of the function is |a|.
• the number of cycles in the interval from 0 to 2π
radians is b.
• the period of the function is 2≠ .
b
The graphs of the cosine (blue) and sine (purple) functions are defined for
θ values that are negative and for θ values that are larger than 360°. The
measure of an angle is positive when the rotation is counterclockwise. The
measure of an angle is greater than 360° when more than one full rotation of
the unit circle has been completed. Every angle that is negative or is greater
than 360° is in the same position as an angle between 0° and 360°. Thus, the
sine curve repeats itself as θ → ∞ and as θ → –∞.
456 Chapter 10 Trigonometric Functions and Identities
Example 2 Graphing Cosine Functions
Graph the function y = 2cos θ over the interval 0 ≤ θ ≤ 2π.
Solution
The amplitude of the function is 2 because a = 2. Because b = 1, the
function will complete 1 cycle in the interval 0 ≤ θ ≤ 2π. The period, or
horizontal length, of the function is 2b≠ = 21≠ = 2≠ . Choose appropriate
scales for the vertical and horizontal axes. Then graph the function. The
graph will complete 1 cycle and range from –2 to 2.
Critical Thinking The function y = cos (θ – x) has a graph that is
x radians out of phase with the function y = cos θ. What value of x would
π
result in a graph that is the same as the standard sine curve y = sin θ? x = 2
The Tangent Function
The graph of the function y = tan θ is also a periodic function that repeats its
shape over and over. The graph is centered about the origin, has a period of
π radians or 180°, and vertical asymptotes that repeat every π units.
10.4 Graphing Trigonometric Functions
457
The tangent function behaves asymptotically because y = tan θ is
undefined when θ = π ± aπ where a is an integer and a ≠ 0. Recall that
2
y
tan θ = = sin θ and is therefore undefined whenever cos θ = 0.
x cosθ
Characteristics of Tangent Functions
The generalized tangent function y = atan bθ, with
a ≠ 0, b > 0, and θ in radians has the following
properties:
• one cycle occurs in the interval – ≠ to ≠ .
2b
2b
• the period of the function is ≠ .
b
• there are vertical asymptotes between each repeated
cycle of the graph.
Example 3 Graphing Tangent Functions
Graph the function y = tan πθ over the interval –1.5 ≤ θ ≤ 1.5.
Solution
Because b = π, the period of the function is ≠
≠ =1 and one cycle occurs in
π
1
≠
1
the interval − = − to
= . So there will be a vertical asymptote at
2
2π
2≠ 2
x = − 1 , x = 1 , and at each additional unit. Choose appropriate scales for
2
2
the horizontal and vertical axes and graph the function.
Ongoing Assessment
Graph the function y = tan ≠ θ over the interval –6 ≤ θ ≤ 6.
4
458 Chapter 10 Trigonometric Functions and Identities
Activity
Using a Graphing Calculator
A graphing calculator can be used to graph trigonometric functions such
as the sine, cosine, and tangent functions. A graphing calculator will also
graph the reciprocal functions: cosecant, secant, and cotangent.
1 Enter y = 1/sin x in the function list on a graphing calculator. Use
the ZTRIG key to set up horizontal axes that are spaced every 90°
or ≠ radians. Graph the function.
2
2 Delete the function from Step 1 and enter y = 1/cos x in the
function list. Then graph the function.
3 Delete the function from Step 2 and enter y = 1/tan x in the
function list. Then graph the function.
4 Recall that these functions are the reciprocals of the basic
trigonometric functions. For example, csc x = 1 . Clear the
sin x
calculator screen and plot the graphs of y = sin x and
y = csc x at the same time. What do you notice about the
cosecant graph at the points where y = 0 in the sine graph?
Explain why this occurs.
Lesson Assessment
Think and Discuss
1. What is the range of the sine function, y = sin θ? What is the
amplitude of the function?
2. How many degrees is it before the graph of the cosine function
begins to repeat? How many radians?
3. Compare and contrast the graphs of the sine function and the
cosine function.
4. What happens to the graph of y = sin θ if the equation is
changed to y = 2sin θ?
5. What happens to the graph of y = tan θ if the equation is
changed to y = tan 2θ?
10.4 Graphing Trigonometric Functions
459
Practice and Problem Solving
Identify the amplitude, period, and number of cycles in the
interval from 0 to 2π for each trigonometric function. (Assume
that θ is given in radians.)
6. y = 1.5sin 2θ
8. y = –2sin θ
10. y = 0.2cos 0.2θ
12. y = 3sin (θ + π)
7. y = 4cos θ
9. y = 3cos θ
2
11. y = –1.5sin θ
3
θ
13. y = cos ( − π )
2 2
1
2
1
3
1
2
Identify the period and the location of two vertical asymptotes
for each function. (Assume that θ is given in radians.)
14. y = tan 2θ 15. y = 3tan θ
16. y = 7.5tan θ 2
17. y = 2tan (θ + π)
Graph each trigonometric function over the specified domain.
18. y = 2cos 2θ, 0 ≤ θ ≤ 2π
20. y = tan 2θ, –180° ≤ θ ≤ 180° 21. y = 2cos θ, –2π ≤ θ ≤ 2π
22. y = tan
24. Sounds traveling through the air can be modeled by sine waves.
The sound of a car horn can be represented by the function
y = 8sin 160θ.
≠
θ, –4.5 ≤ θ ≤ 4.5
3
19. y = sin 2θ, 0 ≤ θ ≤ 360°
23. y = –1.5sin θ, –2π ≤ θ ≤ 2π
a. The loudness of a sound is related to the
amplitude of its sound wave. Write a new
function for a sound that is twice as loud as
the car horn.
b. Sounds are caused by vibrations. When an
object vibrates twice as fast, it produces
a sound with a pitch that is one octave
higher. The period of the sound wave is half
the original period. Write a new function
for a sound that is the same loudness as
the car horn, but one octave higher.
c. Graph the function modeling the sound of the car horn over
the interval 0 ≤ θ ≤ 0.1.
460 Chapter 10 Trigonometric Functions and Identities
25. The function y = 3sin ≠ t models the position of a fishing boat on
4
the ocean relative to sea level. The independent variable t is given
in seconds, and the dependent variable y is given in feet above or
below sea level.
a. Graph the trigonometric function for values of θ between
0 and 12 seconds.
b. How long does it take the boat to complete one full cycle as it
rides the waves of the ocean?
c. What is the amplitude of the function? What is the physical
interpretation of this value?
d. How would the equation of the function change if the boat
were riding on a 4-foot wave? y = 4 sin π t
4
Mixed Review
Determine if each sequence is arithmetic. If so, state the
common difference of the sequence.
26. 4, 12, 20, 28, 36, …
27. 1, 2, 4, 7, 11, …
Find the area of each triangle. Round your answers to the
nearest tenth if necessary.
28.
29.
30. A cereal box has a width of 2x inches, a length of 4x + 1 inches,
and a height of 6x + 1 inches. Write an expression for the
volume of the box. Then find the volume of the cereal box if
x = 1.5 inches.
10.4 Graphing Trigonometric Functions
461
Lesson 10.5 The Law of Sines
and Law of Cosines
Objectives
Use the Law of Sines to
solve for missing sides
and/or angle measures
in triangles.
Find the area of triangles.
The sine ratio is used to find missing side lengths in right triangles when
at least one angle measure and one side length is known. However, this
ratio cannot be used in triangles that are not right triangles.
For any triangle ABC, if a, b, and c represent the side lengths opposite
angles A, B, and C, respectively, then the Law of Sines states that
sin
A sin
B sin C .
=
=
a
b
c
Finding Side Lengths Using the Law of Sines
The Law of Sines can be used to find missing side lengths
of triangles. It can be used to find missing side lengths in both right
triangles and non-right triangles when two of the angle measures and
at least one of the side lengths is known.
Example 1
Finding the Side Length
of a Triangle
In LMN, m∠L = 59°, m∠M = 45°, and MN = 12 yards. Find the
length of LM to the nearest tenth.
Solution
Draw and label LMN.
Find the measure of ∠N.
m∠N = 180° – 59° – 45° = 76°
Use the Law of Sines to solve for LM.
sin 59° sin 76°
=
LM
12
12 sin 76°
LM =
sin 59°
LM ≈ 13.6
462 Chapter 10 Trigonometric Functions and Identities
Side LM is about 13.6 yards long.
Ongoing Assessment
In PQR, m∠P = 35°, m∠R = 78°, and PQ = 9.5 in. Find the length of PR
to the nearest tenth. 8.9 in.
Finding Angle Measures Using the Law of Sines
The Law of Sines can also be used to find missing angle measures in
right triangles and non-right triangles. Sufficient information must be
provided about the triangle so that a proportion with only one variable can
be formulated.
Example 2 Finding the Angle Measure of a Triangle
In RST, m∠R = 62°, RS = 14 meters, and ST = 16 meters. Find m∠T to
the nearest tenth.
Solution
Draw and label LMN.
Use the Law of Sines to solve for m∠T.
sin 62° sin x°
=
16
14
14sin 62°
= sin x°
16
 14sin 62° 
−1
sin −1 
 = sin (sin x°)
16


50.6° ≈ x
The measure of ∠T is approximately 50.6°.
Ongoing Assessment
In MNO, m∠M = 83°, MN = 8 in., and NO = 10 in. Find m∠O to the
nearest tenth. 52.6°
Critical Thinking In Example 2, how could m∠S be found? After finding
m∠T, subtract m∠R and m∠T from 180° to find m∠S.
10.5 The Law of Sines and Law of Cosines
463
Finding Area Using the Law of Sines
An oblique triangle is any triangle that does not contain a right angle. In
an oblique triangle with sides a, b, and c, the height of the triangle can be
found by h = csin A, h = asin B, or h = bsin C. So the area, K, of the triangle
can be found using the equations K = 0.5(bc)(sin A), K = 0.5(ac)(sin B), or
K = 0.5(ab)(sin C).
Example 3 Finding the Area of a Triangle
Find the area of ABC. Round to the nearest tenth.
Solution
Use the expression for the area of an oblique triangle. In the triangle, c = 22,
a = 10, and m∠B = 25°.
K = 0.5(ac)(sin B)
K = 0.5(10)(22)(sin 25°)
K ≈ 46.5
The area of ABC, rounded to the nearest tenth, is 46.5 ft2.
Ongoing Assessment
Find the area of XYZ. Round to the
nearest tenth. 87.6 cm2
464 Chapter 10 Trigonometric Functions and Identities
Activity
Ambiguous Cases
Two triangles that have a
congruent angle and two pairs
of congruent sides are not
necessarily congruent as shown
below. The Law of Sines can be
used to find the missing angle
measures for both triangles.
1 Use the Law of Sines to express the relationship between
∠A and ∠B. sin 43° = sin B
16
20
2 Solve the equation from Step 1 for m∠B. Round your answer to
the nearest tenth. 58.5°
3 The sine function is positive in Quadrant I and II. Therefore,
another value for m∠B can be found by subtracting the answer
from Step 2 from 180°. Find another value for m∠B. 121.5°
4 Find the other missing angle measures in the two triangles.
Express the angles of the two triangles that satisfy the given
conditions and the Law of Sines. Blue triangle: 43°, 58.5°, 78.5°;
Orange triangle: 43°, 121.5°, 15.5°
For any triangle ABC, if a, b, and c represent the side lengths opposite
angles A, B, and C, respectively, then the Law of Cosines states that
a2 = b2 + c2 – 2bccos A,
b2 = a2 + c2 – 2accos B, and
c2 = a2 + b2 – 2abcos C.
The Law of Cosines can be used to find missing measurements in triangles
when the measures of two sides and the angle between them are known or
when the measures of all three sides are known.
10.5 The Law of Sines and Law of Cosines
465
Finding Side Lengths Using the Law of Cosines
Example 4 Finding the Side Length of a Triangle
Reggie wants to know the distance between the items located at B and C
for the scavenger hunt. If AB = 150 yards, AC = 220 yards, and m∠A = 48°,
what is the distance between the items located at B and C?
Solution
Sketch the locations of the scavenger hunt items to model the problem.
Use the Law of Cosines to solve for BC. The measures of ∠A, b, and c are
known. Solve for a.
a2=b2 + c2 – 2bccos A
a2=(220)2 + (150)2 – 2(220)(150)(cos 48°)
a2≈26,737.4
a≈163.5
The distance between the items is approximately 163.5 yards.
Finding Angle Measures
Example 5 Finding the Angle Measures
of a Triangle
Find m∠A in the triangle. Round your answer to the nearest tenth.
Solution
The measure of ∠A cannot be solved directly. The Law of Cosines can be
used to solve for b, the length of side AC.
b2=a2 + c2 – 2accos B
b2=(30)2 + (15)2 – 2(30)(15)(cos 75°)
b2≈892.1
b≈29.9
Now the lengths a, b, c, and m∠B are known. Use the Law of Sines to solve
for m∠A.
sin 75ϒ = sin A
29.9
30
0.96916≈sin A
–1
sin 0.96916≈sin–1 (sin A)
75.7°≈A
The measure of ∠ A, rounded to the nearest tenth, is 75.7°.
466 Chapter 10 Trigonometric Functions and Identities
Lesson Assessment
Think and Discuss
see margin
1. Explain how the Law of Sines relates the angles of a triangle to
the sides of the triangle.
2. Two angle measures of a triangle and the length of a side
opposite one of the angles are known. Explain how to find the
unknown angle measure and the other side lengths.
3. Is it possible to use the Law of Sines if you only know the three
side lengths of a triangle? Is it possible to use the Law of Cosines
if you only know the three side lengths of a triangle? Explain.
4. In triangle ABC, what happens to the Law of Cosines if
m∠C = 90º? Explain.
Practice and Problem Solving
Use triangle ABC to find the indicated side length. Round your
answers to the nearest tenth if necessary.
5.Find a if m∠A = 78º, m∠C = 63º, and c = 12. 13.2
6.Find b if m∠B = 55º, m∠A = 80º, and a = 7.6. 6.3
7.Find c if m∠A = 74º, m∠C = 65º, and a = 22. 20.7
8.Find b if m∠B = 60º, m∠C = 50º, and c = 30. 33.9
Use triangle RST to find the indicated angle measure. Round
your answers to the nearest tenth if necessary.
9. Find m∠R if s = 13, r = 9, and m∠S = 52º. 33.1º
10. Find m∠T if r = 10, t = 6, and m∠R = 65º. 32.9º
11. Find m∠S if r = 20, s = 14, and m∠R = 80º. 43.6º
12. Find m∠T if s = 6.1, t = 4.3, and m∠S = 12.5º. 8.8º
10.5 The Law of Sines and Law of Cosines
467
Use triangle ABC to find the indicated side length. Round your
answers to the nearest tenth if necessary.
13.Find a if m∠ A = 81º, b = 10, and c = 8. 11.8
14.Find b if m∠B = 64º, a = 7, and c = 4. 6.4
15.Find c if m∠C = 29º, a = 9, and b = 7. 4.4
16.Find b if m∠B = 50º, a = 10.4, and c = 16.7. 12.8
Use triangle ABC to find each indicated angle measure. Round
your answers to the nearest tenth if necessary.
17. Find m∠B if a = 13, b = 9, and c = 8. 43.0º
18. Find m∠ A if a = 6, b = 4, and c = 8. 46.6º
19. Find m∠C if a = 18, b = 15, and c = 12. 41.4º
20. Find m∠ A if a = 4.6, b = 7.1, and c = 10.2. 23.0º
21. A hiking group built a bridge across a ravine from point P to
point Q. A diagram is shown at the left. What is the length, x, of
the bridge? Round your answer to the nearest tenth if necessary.
25.7 yd
22. Use the Law of Sines to estimate the distance, d, across the river.
Round your answer to the nearest tenth if necessary. 41.5 m
468 Chapter 10 Trigonometric Functions and Identities
23. Two points in a forest, A and B, are separated by a lake.
A forest ranger draws a baseline AC on one side of the lake so
that AC = 140 meters. He then uses surveying equipment to
determine that m∠A = 72° and m∠C = 50°. What is the distance
from point A to B across the lake? Round your answer to the
nearest tenth if necessary. 126.5 m
24. The Great Pyramid at Giza has four triangular faces with the
dimensions shown. What is the area of one of the pyramid’s
faces? Round your answer to the nearest square meter. 16,809 m2
25. A boat racing course is shaped like a triangle with a
36º angle between two legs that measure 5.2 kilometers and
4.9 kilometers. What is the length of the third leg of the course?
Round your answer to the nearest tenth if necessary.
3.1 km
26. The lengths of the sides of a triangular sheet of metal are
4.5 inches, 5.2 inches, and 3.8 inches. What is the measure of
the largest angle of the sheet of metal? Round your answer to the
nearest tenth if necessary. 77.1º
27. How many square feet of material were used to build the front
face of the tent shown below? Round your answer to the nearest
tenth if necessary. 78.1 ft2
10.5 The Law of Sines and Law of Cosines
469
28. A surveyor wants to estimate the height of a mountain peak. She
locates two observation points, A and B, 4,456 yards apart at the
base of the mountain. From point A, the angle of elevation to
the peak of the mountain, C, is 23º. From point B, the angle of
elevation is 35º. What is the height of the mountain peak to the
nearest yard? 4,803 yd
29. Two tanker ships leave a port at the same time. The first ship
plots a course 28º north of due west and travels at a rate of
12.6 nautical miles per hour. The second ship plots a course
7º south of due west and travels at a rate of 10.2 nautical
miles per hour.
a. How far has each ship traveled after 2.5 hours? Sketch a
diagram to model their positions after 2.5 hours.
31.5 nautical miles and 25.5 nautical miles; see margin for diagram
b. What is the distance between the two tanker ships after
2.5 hours? Round your answer to the nearest tenth.
18.1 nautical miles
Mixed Review
Find all possible rational roots using the Rational Root Theorem.
30. x3 – 4x2 + 2x + 3 ±3, ±1
31.2x4 + 8x3 – x2 – x + 12 ±12, ±6, ±4, ±3, ±2, ±1
32. The wheels on Jason’s bicycle have a diameter of 20 inches. There
are several spokes evenly spaced along the wheel. If the central
angle between each pair of neighboring spokes is 22.5°, what is
the arc length of the wheel between each pair of spokes? Round
to the nearest tenth. 3.9 in.
470 Chapter 10 Trigonometric Functions and Identities
Lesson 10.6 Verifying Trigonometric
Identities
Objectives
Simplify a
trigonometric
expression.
Verify a
trigonometric
identity.
Trigonometric Identities
The following trigonometric identities can be used to simplify
expressions. These are equations that are true for all values of θ for
which the expressions on each side of the equation are defined.
Reciprocal Identities
csc θ = 1 sec θ = 1 sin θ
cos θ
cot θ =
1
tan θ
Tangent and Cotangent Identities
tan θ =
sin θ cos θ
cot θ =
cos θ
sin θ
Pythagorean Identities
cos2 θ + sin2 θ = 1 1+ tan2 θ = sec2 θ
1+ cot2 θ = csc2 θ
Example 1 Simplifying a Trigonometric Expression
Simplify the expression cos θ + sin θ tan θ.
Solution
Use the identity tan θ =
sin θ
.
cos θ
cos θ + sin θ tan θ
 sin θ 
cos θ + sin θ 

 cos θ 
2
cos θ + sin θ
cos θ
 1
Multiply the expression by 1 in the form of 
 cos θ
2
 1 

sin θ
cos θ 
  cos θ + cos θ 
 cos θ  

1 (cos 2 θ + sin 2 θ )
cos θ

 cos θ .

Because of the Pythagorean Identity cos2 θ + sin2 θ = 1, the result is
1 = sec θ.
cos θ
Ongoing Assessment
Simplify the expression sin θ sec θ cot θ. 1
10.6 Verifying Trigonometric Identities
471
Trigonometric expressions can also be simplified to verify trigonometric
identities. When verifying an identity, you should transform one side of the
equation until it is identical to the other side of the equation. Although there
are no universal strategies for verifying trigonometric identities, it is often
helpful to transform the more complicated side into the less complicated side.
Example 2 Verifying a Trigonometric Identity
Verify the Pythagorean Identity 1 + cot2 θ = csc2 θ.
Solution
Use the identity cot θ =
cos θ
to rewrite the Pythagorean Identity.
sin θ
 cos θ 
1 + cot 2 θ = 1 + 

 sin θ 
2
= 1 + cos2 θ
sin θ
2
Multiply the expression by 1 in the form of sin 2 θ
(
)(
1
cos 2 θ
= sin 2 θ
1
+
sin 2 θ
sin 2 θ
= 12 (sin 2 θ + cos 2 θ )
sin θ
= 12 (1)
sin θ
= csc 2 θ
)
( sin1 θ ) and simplify.
2
Ongoing Assessment
Verify the Pythagorean Identity 1 + tan2 θ = sec2 θ. see margin
Activity 1 Proving a Pythagorean Identity
Prove cos2θ + sin2θ = 1. see margin
1 The Pythagorean Theorem states that a2 + b2 = c2, which can
be written as opposite2 + adjacent2 = hypotenuse2. Divide the
equation opposite2 + adjacent2 = hypotenuse2 by hypotenuse2.
2 Simplify your answer to Step 1. Recall, sine =
cosine =
adjacent
.
hypotenuse
472 Chapter 10 Trigonometric Functions and Identities
opposite
and
hypotenuse
Activity 2 Using the Pythagorean Identity
to Find Trigonometric Values
Let cosθ = 4 and θ be in quadrant IV. Find sinθ and tanθ.
5
1 Complete the table using in the Pythagorean Identity.
Step
Reason
cos2 θ + sin2 θ = 1
Pythagorean Identity
2
 4 
  + sin2 θ = 1
 5 
Substitute 4 for cos θ.
5
16
+ sin2 θ = 1
25
Simplify the exponent.
sin2 θ =
9
25
sin θ =
3
5
Subtract
16
from both sides.
25
Take the square root of both sides.
2 Determine whether sine is positive or negative in quadrant IV.
sin is negative in quadrant IV.
3
3 Find sin θ in quadrant IV. sin θ = −
5
4 Find tan θ in quadrant IV. Show your work. see margin
Lesson Assessment
Think and Discuss
see margin
1. Give an example of a trigonometric identity that is equal to 1.
2. How can sin θ be expressed in terms of cos θ?
3. What does it mean for a trigonometric equation to be a
trigonometric identity?
4. Which trigonometric function is the reciprocal of the
sine function?
5. Create a trigonometric identity of your own by beginning with a
simple trigonometric expression and working backward.
10.6 Verifying Trigonometric Identities
473
Practice and Problem Solving
Simplify each expression.
6. 1 – cos2 θ sin2 θ
7.tan θ cos θ sin θ
8.sin2 θ sec θ csc θ tan θ
9.sin2 θ + tan2 θ + cos2 θ sec2 θ
10. (1 + cot2 θ)(sec θ) csc2 θ sec θ
11.sin θ csc θ – cos2 θ sin2 θ
12.sec θ cos θ – cos2 θ sin2 θ
13.sec θ cos2 θ csc θ cot θ
Verify each identity. check students’ work; see margin for samples
14.cos θ tan θ = sin θ
15.tan θ (cot θ + tan θ) = sec2 θ
16.sin θ sec θ = tan θ
17.cos θ + sin θ tan θ = sec θ
18.sec2 θ = 1 + tan2 θ
19.sec θ – sin θ tan θ = cos θ
20.cos θ sin θ (cot θ + tan θ) = 1
21.sin2 θ tan2 θ = tan2 θ – sin2 θ
7
22. Let sin θ = −
and θ be in quadrant III. see margin
25
a. Find cos θ.
b. Find tan θ.
15
23. Let tan θ = −
and θ be in quadrant II. see margin
8
a. Find sin θ.
b. Find cos θ.
24. How can you express tan θ in terms of cos θ? tan θ =
± 1− cos2 θ
cos θ
25. How can you express cot θ in terms of csc θ? cot θ =
± csc2 θ − 1
26. How can you express sec θ in terms of tan θ? sec θ =
± 1+ tan2 θ
27. Mario simplified the
trigonometric expression
sin2 θ sec2 θ + 1 as
shown here. What error
did he make? Simplify
the expression. see margin
28. Verify the trigonometric identity csc θ tan θ = sec θ.
2
2
29. Simplify the trigonometric expression cot2 θ − csc 2 θ . 1
tan θ − sec θ
sin 2 θ sec 2 θ + 1 = sin 2 θ 12 + 1
sin θ
=1+1
=2
check students’ work
Mixed Review
Write a recursive formula for each sequence.
30. 7, 2, –3, –8, –13, … an
an–1
a1
474 Chapter 10 Trigonometric Functions and Identities
31. 2, –5, 16, –47, 142, …
an
an–1
a1
Lesson 10.7 Angle Sum and Difference
Identities
Objectives
Find exact values
of trigonometric
expressions using
sum and difference
identities.
Angle Sum and Difference Identities
The tables below summarize the Angle Sum Identities and the
Angle Difference Identities. Each of the Angle Difference Identities
can be verified using Angle Sum Identities such as sin (A + (–B)).
Angle Sum Identities
• sin(A + B) = sin A cos B + cos A sin B
• cos(A + B) = cos A cos B – sin A sin B
• tan(A + B) = tan A + tan B
1− tan A tan B
Angle Difference Identities
• sin(A – B) = sin A cos B – cos A sin B
• cos(A – B) = cos A cos B + sin A sin B
• tan(A – B) = tan A − tan B
1+ tan A tan B
Example 1 Using Angle Sum and Difference Identities
Find the exact value of tan 135°.
Solution
It is known that tan 180° = 0 and tan 45° = 1. Because
180° – 45° = 135°, the tangent difference identity can be used to find
the exact value of tan 135°.
tan( A − B ) = tan A − tan B
1 + tan A tan B
tan(180° − 45°) = tan 180° − tan 45°
1 + (tan 180°)(tan 45°)
=
0 −1
1 + (0)(1)
= −1
1
= −1
The exact value of tan 135° is –1.
10.7 Angle Sum and Difference Identities
475
Ongoing Assessment
Find the exact value of cos 240°.
−1
2
Example 2 Using Angle Sum and
Difference Identities
Find the value of the expression sin 110° cos 65° – cos 110° sin 65°.
Solution
Use the angle difference identity sin(A – B) = sin A cos B – cos A sin B to
evaluate the expression.
sin(A – B) = sin A cos B – cos A sin B
sin(110° – 65°) = sin 110° cos 65° – cos 110° sin 65°
sin(110° – 65°) = sin 45° =
2
2
So sin 110° cos 65° – cos 110° sin 65° =
2.
2
Ongoing Assessment
Find the value of the expression sin 160° cos 110° + cos 160° sin 110°. –1
Lesson Assessment
Think and Discuss
see margin
1. Is sin (A + B) = sin A + sin B? If not, give a counterexample.
2. Is cos (A + B) = cos A + cos B? If not, give a counterexample.
3. What expression is equal to sin (A – B)?
4. Explain how to use a difference identity to find the exact value of
sin 142° cos 112° – cos 142° sin 112°.
5. Explain how to use mental math and a sum identity to find the
exact value of cos 71° cos 19° – sin 71° sin 19°.
476 Chapter 10 Trigonometric Functions and Identities
Practice and Problem Solving
Use a sum or difference identity to find the exact value of
each expression.
1
2
6. sin 150°
8. tan 135° –1
7. cos 120° – 21
9. cos 300°
10. tan 225° 1
11. cos 135°
12. tan 15°
13. sin 225°
14. tan 105°
16. cos 33° cos 27° – sin 33° sin 27°
17. sin 156° cos 66° – cos 156° sin 66° 1
18. cos 58° cos 13° + sin 58° sin 13°
19. sin 22° cos 8° + cos 22° sin 8°
tan 31° + tan 14°
20. 1− tan 31° tan 14° 1
2− 3 − 3 −2
15. sin 390°
1
2
2
2
2
−
2
1
2
−
1
2
2
2
1
2
Write each expression as a trigonometric function of a single
angle measure.
21. sin 3θ cos 2θ + cos 3θ sin 2θ sin 5θ
22. cos 4θ cos 2θ – sin 4θ sin 2θ cos 6θ
23. sin 2θ cos θ – cos 2θ sin θ sin θ
24. cos 3θ cos θ + sin 3θ sin θ cos 2θ
tan 3θ − tan θ
25.
tan 2θ
1+tan 3θ tan θ
26. Use a graphing calculator to graph the function y = sin x and
y = (sin x + 30) on the same coordinate grid. Use the interval
0° ≤ x ≤ 360°.
a. Describe the graphs of the functions. see margin
b. For what value(s) of x in the interval 0° ≤ x ≤ 360° are the
functions equal? x = 75°, 255°
10.7 Angle Sum and Difference Identities
477
27. The figure at the right shows
a Ferris Wheel with a radius of
26 feet. Point A represents a
30° rotation of the wheel from
its starting position. Point B
represents an additional θ°
rotation of the Ferris Wheel.
a.The x-coordinate of point B
is 26 cos (θ + 30°). Use an
angle sum identity to write
this expression in terms of
cos θ and sin θ. 13 3 cos θ – 13 sin θ
b.The y-coordinate of point B
is 26 sin (θ + 30°). Use an
angle sum identity to write this
expression in terms of cos θ
and sin θ. 13 3 sin θ + 13 cos θ
c. What are the coordinates of
point B if θ = 60°? (0, 26)
d. What are the coordinates of
point B if θ = 150°? (–26, 0)
Mixed Review
Find the constant of variation, k, in each joint variation. Then
write the joint variation equation.
28. z = 30 when x = –3 and y = 5 k = –2, z = –2xy
29. z = 36 when x = 2 and y = 12 k = 1.5, z = 1.5xy
30. The city hall custodial staff arranged 26 chairs in the first row
of a meeting room for a town meeting. Each additional row
had 2 more chairs than the previous row. There were 12 rows
altogether. How many chairs are in the meeting room? 444
478 Chapter 10 Trigonometric Functions and Identities
Lesson 13.4 Double-Angle and
Half-Angle Identities
Objectives
Find exact values of
trigonometric expressions
using double-angle and
half-angle identities.
A radio wave is an electromagnetic wave
transmitted by an antenna. Radio waves
have different frequencies. When tuning
a radio receiver to a specific frequency, a
specific signal can be picked up. Listening
to a radio station, such as 99.1 FM,
“The Wave,” means that a radio station
is broadcasting an FM radio signal at a
frequency of 99.1 megahertz. Megahertz
means “millions of cycles per second.”
So “99.1 megahertz” means that the
transmitter at the radio station is oscillating
at a frequency of 99,100,000 cycles per
second. A transmitter at the radio station
needs to evaluate sin 150° to isolate a radio
frequency. What is the isolated frequency?
Double-Angle Identities
The Double Angle Identities are special cases of the Angle Sum
Identities in which A = B. If A = B, then the cos (A + A) can be
expressed as cos 2A.
Therefore, cos (A + A) = cos A cos A – sin A sin A
= cos2 A – sin2 A
Note that the Double-Angle Identity for cos has multiple variations.
These variations can be derived using trigonometric identities.
The table below summarizes the Double-Angle Identities.
Double-Angle Identities
• cos 2θ = cos2 θ – sin2 θ
• cos 2θ = 2cos2 θ – 1
• cos 2θ = 1 – 2sin2 θ
• sin 2θ = 2sin θ cos θ
• tan 2θ = 2tan θ
1− tan2 θ
10.8 Double-Angle and Half-Angle Identities
479
Example 1 Using the Double-Angle Identity
Use a double-angle identity to find the exact value of cos 240°.
Solution
3 , the identity cos 2θ = 1 – 2 sin2 θ can be used to
2
find the exact value of cos 240°.
Because sin 120° =
cos 2θ=1 – 2 sin2 θ
cos (2 • 120°)=1 – 2(sin 120°)2


cos 240°=1 – 2  3 
2


2
()
cos 240°=1 – 2 3
4
1
cos 240°= −
2
The exact value of cos 240° is − 1 .
2
Ongoing Assessment
Use a double-angle identity to find the exact value of cos 120°.
−1
2
Half-Angle Identities
The table below summarizes the Half-Angle Identities. The Half Angle
Identities can be derived from the Double Angle Identities. As part of
the derivation process, you must take the square root of both sides of the
equation. Recall that you must include ± when taking the square root of both
sides of an equation.
Half-Angle Identities
• sin A = ± 1− cos A
2
2
• cos A = ± 1+ cos A
2
2
• tan A = ± 1− cos A
2
1+ cos A
When using the half-angle identities, choose the sign for each function as
appropriate for the angle. For example, the sine function is positive if the
terminal side of the angle lies in quadrants I or II. Otherwise, it is negative.
The table at the top of the next page summarizes this concept.
480 Chapter 10 Trigonometric Functions and Identities
When Trigonometric Function Values Are Positive
• The sine function is positive for angles in
Quadrants I and II.
• The cosine function is positive for angles in
Quadrants I and IV.
• The tangent function is positive for angles in
Quadrants I and III.
Example 2 Using the Half-Angle Identity
At the beginning of this lesson, a situation was presented in which a
transmitter at the radio station needed to evaluate sin 150° to isolate a radio
frequency. What is the isolated frequency? Use a half-angle identity to find
the exact value of sin 150°.
Solution
1
Because cos 300° = 2 , the identity sin A = ± 1 − cos A can be used to find
2
2
the exact value of sin 150°.
1 − cos A
sin A = ±
2
2
1 − cos 300°
sin 300° = ±
2
2
Choose the positive square root since sin 150° is positive.
1− 1
2
2
1
300
°
=± 2
sin
2
2
sin 300° = ± 1
2
4
sin 300° = 1
2
2
sin 300° = ±
2
The exact value of sin 150° is 1 . So the isolated frequency is 1 .
2
2
Ongoing Assessment
Use a half-angle identity to find the exact value of tan 150°.
−
3
3
10.8 Double-Angle and Half-Angle Identities
481
ActivityArea
The front face of a tent has the measurements shown below. The area
of the face can be found using the formula A = 128sin 60°.
1 Use the double-angle identity sin 2θ = 2sin θ cos θ to find the
exact value of sin 60°. 3
2
2 Write an expression for the exact area of the face of the tent.
64 3 ft2
3 Use a calculator to approximate the area of the face of the tent to
the nearest whole number. ≈ 111 ft2
4 Verify the area for the front face of the tent by first finding the
altitude for the triangle and then using the formula for the area
of a triangle. see margin
5 Suppose the front face of the tent was 20 feet long. What would
be the surface area for the exposed portion of the tent?
about 862 ft2
Lesson Assessment
Think and Discuss
see margin
1. Is sin 2A = sin A + sin A? If not, give a counterexample.
2. Is cos 2A = cos A + cos A? If not, give a counterexample.
3. In which quadrant(s) is the tangent function positive?
4. In which quadrant(s) is the cosine function positive?
482 Chapter 10 Trigonometric Functions and Identities
Practice and Problem Solving
Use a double-angle or half-angle identity to find the exact value
of each expression.
−1
2
5. cos 120°
7. cos 90° 0
9. tan 240°
6. sin 90° 1
8. sin 240°
10. sin 15°
3
3
2
2− 3
2
3
2
11. tan 15°
7−4 3
13. tan 300°
− 3
14. cos 15°
15. tan 120°
− 3
16.
17. cos 60°
19. tan 60°
3
20. sin 22.5°
2− 2
2
21. cos 600°
−1
2
22. tan 22.5°
3−2 2
23. cos 240°
−1
2
24. cos 22.5°
2+ 2
2
25. Find cos 2θ if cos θ = − 2 and 90° < θ < 180°. − 17
25
5
7
26. Find cos θ if sin θ = − 24 and 180° < θ < 270°. − 25
25
27. Find tan θ if cos θ = 4 and 270° < θ < 360°. 31
2
5
1
θ
28. Find sin
if cos θ =
and 0° < θ < 90°. 21
2
2
29. The sound waves generated by a vibrating tuning fork
can be modeled by the function y = 2sin θ. If the
tuning fork vibrates twice as fast, the sound
will be one octave higher. This is given by
the function y = 2sin 2θ. Write the
function in terms of θ to model the
higher octave sound waves. y = 4 sin θ cos θ
30. How could the identity for cos A be
2
derived from the identity for cos 2A?
Show your work. see margin
1
2
12. sin 120°
−
18.
2+ 3
2
3
sin 60° 2
3
cos 150° − 2
10.8 Double-Angle and Half-Angle Identities
483
31. How could the identity for sin A be derived from the identity for
2
cos A ? Show your work. see margin
2
32. When an arrow is shot from a bow,
the distance that the arrow travels is
a function of the initial velocity and
the angle of elevation, θ. The distance
can be modeled by the function
(v )2
d = 0 sin θ cos θ , where v0 is the
16
initial velocity of the arrow, in feet per
second.
a. Use a double angle formula to
rewrite the function in terms of the
(v0 )2
double angle, 2θ. d = 32
sin 2θ
b. What angle of elevation, θ, will
maximize the distance the arrow
travels for a given initial velocity?
Explain. see margin
c. An arrow is shot with an angle of
elevation of 30° and an initial velocity
of 80 feet per second. How many
feet will the arrow travel? Round your answer to the nearest
tenth. 173.2 ft
Mixed Review
33. In ΔPRQ, m∠P = 72°, RQ = 7 units, and PQ = 4.3 units. Find
m∠R to the nearest tenth. 35.7°
34. What is the height, h, of the attic in Jeremy’s house? Round your
answer to the nearest tenth if necessary. 12.7 ft
35. Verify the trigonometric identity
csc θ cos θ
= 1.
cot θ
check students’ work; see margin for sample
484 Chapter 10 Trigonometric Functions and Identities
Lesson 10.9 Solving Trigonometric
Equations
Objectives
Solve trigonometric
equations.
While trigonometric identities are true for all values of θ, most
trigonometric equations are true for only certain values of θ.
Trigonometric equations can be solved by collecting like terms,
finding square roots, factoring, and substitution.
Collecting Like Terms
Example 1 Solving by Collecting Like Terms
Solve 2cos θ – 1 + 3cos θ = cos θ for 0 ≤ θ < 2π.
Solution
Collect like terms to isolate cos θ on one side of the equation.
2cos θ – 1 + 3cos θ=cos θ
2cos θ – 1 + 1 + 3cos θ – cos θ=cos θ – cos θ + 1
4cos θ=1
cos θ= 1
4
Use the inverse cosine function and a graphing calculator to solve
for θ.
cos–1(cos θ)=cos–1 1
4
θ≈1.32
Because the cosine function is also positive in the fourth quadrant,
2π – 1.32 ≈ 4.97 radians is also a solution to the trigonometric
equation. The two solutions for 0 ≤ θ ≤ 2π are 1.32 and 4.97 radians.
Ongoing Assessment
Solve 4tan θ – 3 = 2tan θ – 2 for 0 ≤ θ < 2π.
Finding Square Roots
Example 2 Solving by Square Roots
Solve 4sin2 θ – 1 = 0 for 0 ≤ θ < 2π.
10.9 Solving Trigonometric Equations
485
Solution
Add 1 to both sides of the equation and then divide both sides by 4.
4sin2 θ – 1 + 1=0 + 1
4sin2 θ=1
1
4
1
(sin θ)2=
4
sin2 θ=
Find the square root of both sides of the equation.
1
4
sin θ = ± 1
2
Use the inverse sine function and a graphing calculator to solve for θ.
sin θ =
sin–1(sin θ)= sin −1 1
2
θ≈0.52
Because the sine function is also positive in the second quadrant,
π – 0.52 ≈ 2.62 radians is also a solution to the trigonometric equation. The
values 0.52 + π ≈ 3.66 and 2.62 + π ≈ 5.76 are also solutions to the equation.
So the solutions for 0 ≤ θ ≤ 2π are 0.52, 2.62, 3.66, and 5.76 radians.
Ongoing Assessment
Solve –27cos2 θ + 3 = 0 for 0 ≤ θ < 2π.
Factoring
Example 3 Solving by Factoring
Solve 4cos θ sin θ + 2sin θ = 0 for 0 ≤ θ < 2π.
Solution
Divide both sides of the equation by 2 and then factor sin θ from each term.
4cos θ sin θ + 2sin θ=0
2cos θ sin θ + sin θ=0
sin θ (2cos θ + 1)=0
486 Chapter 10 Trigonometric Functions and Identities
Use the Zero-Product Property to set each factor equal to 0 and solve for θ.
sin θ=0
θ = 0 or θ=π
2cos θ + 1=0
cos θ= − 1
2
2≠
4≠
θ= 3 or θ = 3
4≠
The four solutions of the equation are 0, π, 2≠ , and 3 radians.
3
Ongoing Assessment
Solve 3cos θ sin θ + 3sin θ = 0 for 0 ≤ θ < 2π.
Quadratic Trigonometric Equations
Trigonometric equations can take the form of quadratic trinomials which
can be factored into two binomials. It is helpful to use a substitution process
to simplify the expression prior to factoring. In other words use x in place
of the trigonometric function. Once the expression has been factored, the
substitution process is reversed.
Example 4 Solving Trigonometric Equations
of Degree Two
Solve tan2 θ – 2tan θ – 3 = 0 for 0 ≤ θ < 2π.
Solution
Let tan θ = x.
x2 – 2x – 3=0
(x – 3)(x + 1)=0
x = 3 or x=–1
Substitute tan θ for x.
tan θ = 3 or tan θ = –1
Solve for θ.
tan θ=3
tan θ=–1
–1
–1
–1
tan (tan θ)=tan (3)tan (tan θ)=tan–1(–1)
θ≈1.25 and 4.39
θ≈–0.7854
Because the domain of θ is 0 ≤ θ < 2π and the tangent function is positive in
quadrants I and III, the second equation has solutions –0.7854 + π ≈ 2.36 and
2.36 + π ≈ 5.50 radians. The solutions are 1.25, 2.36, 4.39, and 5.50 radians.
Ongoing Assessment
1
Solve sin2 θ – 3 sin θ + = 0 for 0 ≤ θ < 2π.
8
4
10.9 Solving Trigonometric Equations
487
As a paddle wheel spins on its axis, the height
of a point on the wheel with respect to the
water level can be modeled by the function
≠t
h(t) = 15 – 18cos 12. The height above
or below the water is given in feet, and t is
the amount of time in seconds. Answer the
following questions to describe the paddle
wheel as it spins on its axis.
a.What is the diameter of the wheel?
b.How high above the surface of the water is
the top of the wheel?
c.How long does it take the wheel to complete
one full revolution?
Step 1 Understand the Problem
Describe the problem situation in your own
words. What information do you need to find?
Step 2 Develop a Plan
Problem-solving strategy: Use a graph.
What are suitable axis limits to graph the
cosine function? How will graphing the
function help you answer the questions in
the problem statement?
Step 3 Carry Out the Plan
Graph the trigonometric function on a graphing
calculator or on grid paper using suitable axes.
Use the graph to find the diameter of the
wheel, the maximum height above the surface
of the water, and the amount of time it takes
to complete a full revolution. Explain how you
found your answers.
Step 4 Check the Results
Check your results to make sure they
seem reasonable.
Lesson Assessment
Think and Discuss
1. Explain how to solve the equation cos2 θ – 1 = 0 for 0 ≤ θ  < 180°.
2. How many solutions are there to the equation 4 tan θ – 0.25 = 0?
Explain.
3. If the sine of an angle is positive and one of the solutions in
the interval 0 ≤ θ < 2π is known, how can the other solution
be found?
4. How could a graphing calculator be used to find the solutions of
a trigonometric equation within a particular domain?
488 Chapter 10 Trigonometric Functions and Identities
Practice and Problem Solving
Solve each equation in the given interval. Round your answers
to the nearest hundredth radian or tenth degree.
5.10cos θ = –5, 0 ≤ θ < π
6.1.5tan θ + 2.5 = 0, 0 ≤ θ < 180°
7.5tan θ = 3 + 2tan θ, 0 ≤ θ < 2π
8.2sin θ –
9.3tan θ = tan θ + 1, 0 ≤ θ < 360°
2 = 0, 0 ≤ θ < 360°
10.tan2 2θ = 1, 0 ≤ θ < π
11.2sin2 θ + 3sin θ – 2 = 0, 0 ≤ θ < 180°
12.(sin θ – 1)(sin θ + 1) = 0, 0 ≤ θ < 2π
13.–15cos2 θ + 8 = 0, 0 ≤ θ < 180°
14.3sin2 θ – 2 = 0, 0 ≤ θ < 2π
15.2sin4 θ + sin2 θ = 0, 0 ≤ θ < 360°
16.tan2 θ + tan θ = 0, 0 ≤ θ < 360°
17.4sin2 θ – 4sin θ + 1 = 0, 0 ≤ θ < 360°
18.2sin2 θ – 3sin θ – 2 = 0, 0 ≤ θ < 2π
19. The number of hours of daylight during the day depends on the
time of year. The hours of daylight, y, can be modeled by the
function y = 14 + 3cos
(
2π
(d
365
)
− 100) , where d is the number
of days since January 1. On which day(s) are there 16.5 hours of
daylight? Round to the nearest whole number.
20. A boat trolling through a no-wake zone of
a lake produces 3-inch waves that repeat
every second. This can be modeled by
the equation y = 3cos 2πt, where y is the
height, in inches, of the wave and t is the
time, in seconds. At what times between
0 and 2 seconds is the height of the wave
equal to 2 inches? Round to the nearest
hundredth of a second.
10.9 Solving Trigonometric Equations
489
21. Use a Pythagorean trigonometric identity to solve the equation
4sin2 θ + 3cos θ – 2 = 0 for 0 ≤ θ < 2π. Round your answer to
the nearest hundredth.
22. The height above ground level of a person riding on a Ferris
Wheel can be modeled by the function h(t) = 32 – 30cos ≠t. In
the function, t is the number of seconds since the ride began,
20and
h(t) is the passenger’s height, in feet, above the ground.
a. What are the first two times, t, that the passenger’s height
above the ground is 2 feet? How long does it take to make
one complete revolution around the ride?
b. When does the passenger first reach the highest point of
the ride? How high above the ground is the passenger at
this point?
23. The voltage V in the power cord of a television
after t seconds can be modeled using the function
V(t) = 120cos (120πt).
a. Find the first time the voltage is equal to
0 volts after turning the television on.
b. What is the maximum voltage in the power
cord at any time? Find all the times at which
the voltage is at its maximum.
c. What is the minimum voltage in the power cord at any time?
Find all the times at which the voltage is at its minimum.
Mixed Review
Convert each angle measure from radians to degrees.
4≠
24. 3
5≠
25.
6
7≠
26.
4
27. The lengths of several picture frames at a department store are
3.5 inches, 4.25 inches, 6 inches, and 8 inches. Do these sizes
represent a geometric sequence? If so, state the common ratio.
490 Chapter 10 Trigonometric Functions and Identities
Math Labs
Activity 1: The Circle of Your City
Problem Statement
Equipment
A circle of latitude is the circle that is formed
by a two-dimensional plane parallel to the
equator intersecting a point of latitude. The
equation used to calculate the circle’s radius
is r = Rcos θ where r is the length of the
radius of the circle of latitude, R is the length
of the radius of Earth (3,960 miles) and θ
is the latitude in degrees. Find the radius
of the circle of latitude for any city given
the city’s latitude, and explain how a city’s
temperature relates to its latitude.
Computer with
Internet
access
Scientific
calculator
Procedure
1 Choose any city in the world. Find a website where you can
determine the city’s latitude. sample answer: Jacksonville, FL;
latitude = 30° 19’ 55”
2 If the latitude has minutes and seconds, then convert it to a
decimal. Round your answer to the nearest hundredth.
sample answer: latitude = 30.33°
3 Find the length of the radius, to the nearest tenth, of the circle
of latitude for your city. sample answer: r = 3,418.0 miles
4 Choose another city in the opposite hemisphere and repeat
Steps 1 through 3. sample answer: Punta Arenas, Chile; r = 2,030.7 miles
5 Choose a city located on the equator. Without using the
equation, find the length of the radius of the circle of latitude.
Verify your answer using the equation. For any city on the equator,
r = 3,960 miles.
6 Draw a circle to represent the Earth. Mark the equator with
its radius. see margin
7 Use an arc to represent the circle of latitude for each of your
chosen cities. Label the radius of each. see margin
8 Write a general statement about the radius of a circle of
latitude and how it relates to the temperature of a city.
see margin
Math Labs
491
Activity 2: The Sun Today
Problem Statement
Equipment
Use a protractor to approximate the angle
of the sun at different times during the
day. Model the position of the sun in the
sky with a graph. Then use your graph to
predict the position of the sun in the sky at
different times during the day.
Clock
Grid paper
Level
Protractor
Yard sticks
Procedure
1 Work with a partner. At various times throughout the day (on
a sunny day) go outside and use the yard sticks, level, and
protractor to measure the angle of the sun with respect to a
level horizon. Have your partner hold a yard stick level in the
direction of the sunrise. Form an angle by pointing the other
yard stick directly at the center of the sun. Hold the yard sticks as
steady as possible and use the protractor to measure the angle.
2 Collect data at various times during the day to complete a
table such as the one shown below. see margin
3 Use your protractor to draw a
large semicircle on grid paper.
Place markings along the
semicircle at every 30° like the
one shown at the left. Draw
each angle from your data on
the semicircle and label each
corresponding time. see margin
4 What was the change in the
Time of
Day
9:00 A.M.
10:00 A.M.
11:00 A.M.
12:00 Noon
1:00 P.M.
2:00 P.M.
3:00 P.M.
Angle of
the Sun
angle of the sun between 9:00 A.M. and 10:00 A.M.? Use
your data and your graph to predict the position of the sun
at 9:30 A.M. sample answer: 15°, about 52.5° off the Eastern horizon
5 What was the change in the angle of the sun between noon and
1:00 P.M.? Use your data to predict the position of the sun at
12:30 P.M. sample answer: 15°, about 97.5° off the Eastern horizon
6 How do you think your data would change if it were gathered
at different times during the year? Answers will vary.
492 Chapter 10 Trigonometric Functions and Identities
Activity 3: The Sine Curve of Biorhythms
Some believe that a person’s physical, emotional, and intellectual
well-being are based on cycles that model sine curves. This
unscientific theory is based on the idea that these three cycles
begin on the day one is born and throughout one’s life the three
cycles have high points and low points that occasionally coincide.
Problem Statement
Equipment
Use the Internet to locate and print a
biorhythm chart based on your date of birth.
Translate each of your biorhythm sine curve
graphs into an equation.
Computer with
Internet
access and
a printer
The equation of a sine curve is given by
y – k = Asin B(x – h) where k is the vertical
offset, h is the horizontal offset, A is the
amplitude and B is the period. Follow the
steps to transform a sine curve graph to an
equation.
Steps to Transform a Sine Curve to an Equation
1. Find the minimum and maximum point on the y-axis and calculate
the amplitude A.
2. Find the period B by using the equation period length = 2≠ where
B
B is the period of the sine curve.
3. Find the first “zero” point. This is the first point where the
horizontal line midway between the minimum and maximum
y values intersects the sine curve. Use this point to calculate k and h.
Procedure
1 Locate a website on the Internet where you can generate a
personalized biorhythm chart. Print a copy of the graphs. A
sample chart is shown on the next page.
2 Find the minimum and maximum values of the Intellectual sine
curve on the y-axis and calculate the amplitude. sample answer:
A = 100
3 Find the period of the Intellectual sine curve. sample answer: B = 2π
4 Estimate the “zero” point and use this coordinate to determine
29
the values of h and k. Sample answer: h = 7, k = 0
Math Labs
493
5 Write the equation in the form y – k = Asin B(x – h) to find the
equation that fits the Intellectual sine curve. see margin
6 Repeat Steps 2 through 5 for the Emotional and Physical
sine curves. see margin
Activity 4: Swing of a Pendulum
Problem Statement
Equipment
Use a string and a washer to create a
pendulum and measure the period and
amplitude of the pendulum. Model the
motion of the pendulum with a periodic
function and graph the function. Experiment
with factors such as the length of the string
and the weight of the pendulum to see how
each affects the swing of the pendulum.
Graphing
calculator
Protractor
Scissors
Stopwatch
String
Washers or
other weights
Procedure
1 Tie one end of a string to a weight such as a washer or a nut.
Cut the string to a length between 8 and 20 inches.
sample string length: 15 in.
2 Attach the other end of the string to a fixed point such as
a coat hanger hook. You could also tie the end to a pencil
and hold the pencil level against a wall to form a swinging
pendulum.
494 Chapter 10 Trigonometric Functions and Identities
3 Hold a protractor upside down so that the 0° vertex is aligned
with the top of the pendulum. Holding the string taut, pull
the weight to one side so that you form a 30° angle with the
string and its vertical resting position. Work with a partner
to time how long it takes the pendulum to swing through
10 complete cycles. sample answer: 12.5 s
4 Time the pendulum as it swings. Find the average amount of
time it takes to complete one full cycle. The average time of
one full cycle is the period for the function.
sample answer: 1.25 s
5 How far from its resting position is the weight at its extremes?
Calculate the arc length of the pendulum along its swing path.
In terms of a periodic function, what is this distance?
sample answer: The amplitude is about 7.85 inches.
6 Write a trigonometric function to describe the distance of the
pendulum from its original resting position t seconds after it
is released. sample answer: y = 7.85 cos 85πt
7 Graph the function over a domain that includes 3 full cycles.
8 Experiment with different lengths of string for the pendulum.
see margin
Compare the period of your pendulum with those of your
classmates. How does changing the length of the string
affect the period of the pendulum? How does it affect
the amplitude? see margin
9 Experiment with different pendulum weights for a fixed length
of string. For example, tie a second weight or a heavier weight
to the end of your string. How does changing the weight of
the pendulum seem to affect the period of the pendulum?
How does it affect the amplitude? see margin
Math Labs
495
Math Applications
The applications that follow are like the ones you will encounter in many workplaces. Use the
mathematics you have learned in this chapter to solve the problems. Wherever possible, use
your calculator to solve the problems that require numerical answers.
General
Content
1 A mounted van ramp is 9 feet long when fully deployed. The
distance between the ground and the point directly above it where
the ramp connects to the van is 2 feet.
a. What is the measure of the angle in radians that the mounted
side-door van ramp makes with the ground? Round to the
nearest tenth of a radian. 0.2 radian
b. What is the measure of the angle in degrees that the mounted
van ramp makes with the ground? Round to the nearest tenth
of a degree. 12.8°
c. If the distance between the ground and the point directly above it where
the ramp connects to the van is 3 feet instead of 2 feet, what will be the
measure of the angle in radians that the mounted van ramp makes with the
ground? Round to the nearest tenth of a radian. 0.3 radian
d. In order to decrease the angle the mounted van ramp makes with the
ground, should the van have a shorter ramp or longer ramp? longer ramp
2 Grace has been chosen to be a contestant on a reality TV show. The show
takes place on a circular island that has two main trails. Trail A runs from
the westernmost point of the island to the easternmost point of the island.
Trail B runs from the northernmost point of the island to the southernmost point
on the island. Both trails are 500 miles long. At the beginning of the show, all
the participants are dropped off at the center of the island. The goal of each
contestant is to be the first person to arrive at the boat dock. The boat dock is
located northwest of the center of the island. The angle formed by the straight
line going from the center of the island directly west and the straight line going
from the center of the island to the boat dock is 11≠ radians.
60
a. How far north of Trail A is the boat dock? Round to the nearest tenth. 136.2 mi
b. How far west of Trail B is the boat dock? Round to the nearest tenth. 209.7 mi
c. If Grace travels along the outermost coast of the island from the
westernmost point of the island to the boat dock, how far will she have to
travel? Round to the nearest hundredth. 143.99 mi
496 Chapter 10 Trigonometric Functions and Identities
3 Cale, Tammie, and Allison are members of the ultimate flying disc team. The
object of the game is to score points by receiving a teammate’s pass of the flying
disc in the opposing team’s end zone. Players may not run when in possession
of the disc. Cale had the disc and passed it to Tammie, who was 18 meters
away. Tammie then passed the disc to Allison in the end zone 28 meters away.
When viewing the instant replay, the statistician calculated the measured angle
formed by the straight line going from Cale to Tammie and the straight line
going from Tammie to Allison to be 75°. Use the facts that sine and cosine of 45°
is 1 , the sine of 30° is 1, and the cosine of 30° is 3 to find cosine 75°.
2
2
2
a. If the straight line distance from Cale and Allison is a and the measure of
the angle formed by the straight line going from Cale to Tammie and the
straight line going from Tammie to Allison is A, use the Law of Cosines to
write a2 in terms of cos A. a2 = 1,108 – 1,008cos A
b. The angle formed by the straight line going from Cale to Tammie and the
straight line going from Tammie to Allison measured 75°. What angle sum
identity can the statistician use to solve for a in the equation given in the
answer to part a? cos (A + B) = cos A cos B – sin A sin B
c. Substitute the values the statistician knows into the angle sum identity
given in the answer to part b. cos (30° + 45°) = 3 • 1 − 1 • 1
2
2
2
2
d. Simplify the answer given to part c to solve for the cosine of 75° to the
nearest hundredth. 0.26
e. Substitute the answer given in part d for cos A in the answer given in part a
and solve for a, which is the straight-line distance from Cale to Allison.
Round to the nearest tenth. 29.1 m
4 A combine is a machine that picks, thrashes, and cleans grain
while traversing a field. Avery has a reclining backrest on her
combine to increase comfort. The measure of the angle formed
by the combine’s seat and the reclining backrest can range from
90° to 135°. The equation tan2 θ – 6tan θ – 16 = 0 represents
this situation where θ is the measure of the angle formed by the
combine’s seat and the reclining backrest. Avery wants to solve for
θ so she can adjust her reclining backrest to this angle in the future.
Agriculture &
Agribusiness
a. What type of trigonometric equation does Avery have to solve
to find θ? quadratic
b. When tan θ is substituted for x, four possible angle measures emerge.
Explain why three of them can not be correct. see margin
c. What is the measure of the angle formed by the reclining backrest and the
seat? Round to the nearest tenth degree. 116.6°
Math Applications
497
5 Juan decided to place a large scarecrow in his rice patty to keep the crows from
ravaging his crop. The scarecrow is 20 feet tall. The angle of the elevation from
the top of the scarecrow’s shadow to the top of the scarecrow has a cosecant
of 1.03.
a. What is the measure of the angle of elevation from the top of the
scarecrow’s shadow to the top of the scarecrow? 75.9°
b. What is the distance from the top of the scarecrow’s shadow to the top of
the scarecrow? Round to the nearest tenth of a foot. 20.6 ft
c. What is the length of the scarecrow’s shadow? Round to the nearest tenth
of a foot. 5.0 ft
Business &
Marketing
6 Ronaldo has a kiosk at the local mall where he sells earrings. One of Ronaldo’s
customers wants to purchase a 24-carat gold-plated triangular earring.
Ronaldo charges $1.75 per mm2 of gold. Two of the sides of the earring have
measurements of 11 millimeters and 12 millimeters, respectively. The angle
between the two sides measures 43°. The customer has $75.
a. What is the area of the 24-carat gold-plated triangular earring that the
customer wants to purchase? Round to the nearest tenth. 45.0 mm2
b. Does the customer have enough money to purchase the earring? no
7 Marissa owns a 24-hour paper-recycling plant. There always exists some paper
waiting to be recycled in the plant. Marissa found the range of the amount of
paper waiting to be recycled is 2 tons, with 4 tons being the most that is ever
waiting to be recycled and 2 tons being the least that is ever waiting to be
recycled. Marissa also found that the amount of paper waiting to be recycled,
in tons, can be represented by a sine curve with an amplitude of 1 and an offset
of 3, and the curve goes through 1 cycle every 360 days. The offset of a sine
curve is the number of units the axis about which the sine curve is oscillating
is above or below 0. On Monday at 12:00 p.m., Marissa noticed the amount of
paper waiting to be recycled was 3 tons and the amount of paper waiting to be
recycled was increasing.
a. How many tons of paper will be waiting to be recycled at 12:00 p.m.
123 days from Monday? Round to the nearest tenth. 3.8 tons
b. How many tons of paper will be waiting to be recycled at 12:00 p.m.
206 days from Monday? Round to the nearest tenth. 2.6 tons
c. How many days from Monday will the amount of paper waiting to be
recycled be the greatest? 90
d. How many days from Monday will the amount of paper waiting to be
recycled be the least? 270
498 Chapter 10 Trigonometric Functions and Identities
8 Bella does freelance marketing work and is designing a logo for a small cafe that
offers free wireless Internet access. The company asked Bella to design a black
circle with a white pie slice, as shown below.
The logo has a circumference of
30π centimeters and the length of the
intercepted arc of the white pie slice is
5π centimeters.
a. What is the measure of the
central angle of the white pie slice
in radians? see margin
b. What is the measure of the central angle of the white pie slice in degrees? 60°
c. If Bella enlarges the logo so that the circumference of the circle is
720π centimeters, will the length of the intercepted arc of the white pie
slice change? Will the central angle of the white pie slice change? yes; no
9 Byron is currently at a convenience store. He is going to drive to the soccer field.
He knows that if it were possible to travel directly in a straight line from the
convenience store to the soccer field, the distance would be 6.5 miles. He also
knows that the angle that is formed by the straight line going directly west and
the straight line going directly to the soccer field measures 30°.
a. Approximately how far should Byron travel west? Round to the
nearest tenth. 5.6 mi
b. Approximately how far should Byron travel north? Round to the
nearest tenth. 3.3 mi
10 Lola decided to rent a canopy for her graduation party. Lola drives three stakes
supporting the canopy into the ground at points A, B, and C. The distance
between A and B is 11 meters. The distance between A and C is 17 meters. The
angle formed by the straight line going from A to B and the straight line going
from A to C is 29°.
a. What type of triangle is formed by the straight line going from A to B, the
straight line going from A to C, and the straight line going from B to C?
oblique
b. What is the area of the triangle that is formed by the straight line going
from A to B, the straight line going from A to C, and the straight line going
from B to C? Round to the nearest tenth. 45.3 m2
Math Applications
499
Family &
Consumer
Science
11 Tamika is taking rowing lessons.
She is learning the term layback,
which is the position of the
upper body at the end of a
rowing stroke. Tamika’s coach
told her at the end of a rowing
stroke, her upper body should
be leaning slightly backward, but
it should not be leaning more
than 10° backward from vertical.
If the measure of the angle at
which Tamika is leaning slightly
backwards is represented by θ,
then 5sin θ cos θ – 4.9sin θ = 0.
Tamika wants to solve for θ so
she knows whether or not she is
following her coach’s instructions.
a. What method can Tamika use to help solve for θ? factoring
b. Is Tamika following her coach’s instructions? no
c. If the measure of the angle at which Tamika is leaning slightly backwards is
represented by θ, and 10sin θ cos θ – 9.9sin θ = 0, is Tamika following her
coach’s instructions? yes
12 The length of the diameter of Anna’s clock is 18 inches. At a given time,
the central angle between the hour hand and the minute hand is 3≠ radians.
20
At a second time the central angle between the hour hand and the minute hand
on the clock is 39°.
a. What is the length of the intercepted arc of the hour hand and the minute
hand at the first time? 27≠ in.
20
b. Which is greater, the central angle between the hour hand and the minute
hand at the first time or the central angle between the hour hand and the
minute hand at the second time? the central angle between the hour hand and the
minute hand at the second time
500 Chapter 10 Trigonometric Functions and Identities
13 Dmitri wants to make sure his keyboard is located at the correct vertical position
to be ergonomically correct. Current guidelines recommend the angle formed
at the elbow by the upper arm and the forearm be between 70° and 135°. If
Health
the measure of the angle formed at Dmitri’s elbow by his upper arm and his
Occupations
forearm when typing is represented by θ, then 45sin θ + cos θ tan θ – 42.5 = 0.
Dmitri wants to calculate the value of θ so he can be sure he is adhering to
current ergonomic guidelines, so he uses a trigonometric identity to simplify the
trigonometric expression so he can solve the equation for θ.
a. What trigonometric identity does Dmitri use to simplify the trigonometric
expression on the left side of the equation? tan θ = sin θ
cos θ
b. What is the simplified trigonometric expression on the left side of the
equation after using the trigonometric identity given in the answer to
part a? 46sin θ – 42.5
c. What is the measure of the angle that should be formed at the elbow by
Dmitri’s upper-arm and the forearm? 112.5°
Industrial
Technology
14 Serena, an astronaut working on the International Space Station, is using a
robotic arm used to move equipment and to service instruments attached to
the station to support astronauts working in space. The arm is attached to the
station at the Mobile Base System (MBS). At the end of the arm is the hand.
While doing some routine maintenance, she notices the arm forms a 36° angle
at 1 of its 7 motorized joints. The segment of the arm going from the MBS
to the joint is 27 feet long. The segment going from the joint to the end of
the hand is 43 feet long. Serena will use the Law of Cosines to calculate the
distance. She knows the sine of 18° is 0.31.
a. What double-angle identity should Serena use to find the distance between
the MBS and the end of the hand? cos 2θ = 1 – 2sin2 θ
b. What is the distance between the MBS and the end of the hand? Round to
the nearest tenth. 26.5 ft
Math Applications
501
Chapter Review
Lessons 10.1 and 10.2
Review Examples
Solve for x. Round to the nearest tenth.
Use the sine ratio to solve for x.
sin 28ϒ= 14
x
x = 14
sin 28ϒ
Use a scientific calculator to evaluate sin 28°.
x ≈ 29.8 in.
Student Review Exercises
Use triangle ABC to find each ratio.
1.cos B
24
25
2.tan A
24
7
Convert each angle measure to degrees
or radians as appropriate. see margin
3.40°
4. 3≠ radians
4
5. The angle from the top of a tree to a
point 60 feet from its base is 44°. What
is the height of the tree? Round the
answer to the nearest tenth. 62.1 ft
Lessons 10.3 and 10.4
Review Examples
Solve for x. Round to the nearest tenth.
Student Review Exercises
Graph each function.
6. y = 2cos θ, 0 ≤ θ ≤ 2π see margin
Use the tangent ratio to write an equation.
4
tan x° =
10
tan x° = 0.4
Use the inverse tangent function and a scientific
calculator to solve for x.
7. y = tan ≠ θ, –4.5 ≤ θ ≤ 4.5 see margin
2
8. A 6-ft board is leaning against a wall
so that the top of the board reaches
a height of 4 feet 9 inches. What is
the measure of the angle formed by
the board and the ground? Round the
answer to the nearest tenth. 52.3º
t an–1 (tan x°)=tan–1 0.4
x=tan–1 0.4
x≈21.8°
502 Chapter 10 Trigonometric Functions and Identities
Lessons 10.5 and 10.6
Review Examples
Find the length, x, of the bridge across
the ravine.
Student Review Exercises
Use the Law of Sines or the Law of
Cosines to find each measure. Round to
the nearest tenth.
9.Find c if m∠C = 35º, a = 6, and b = 8. 4.6
Find the missing angle of the triangle.
180º – 108º – 40º = 32º
Apply the Law of Sines.
10. Find m∠B if a = 18, b = 13, and
m∠A = 67º. 41.7º
11. Verify the trigonometric identity
sin θ tan θ = 1 – cos θ.
cos θ
Check students’ work; see margin for sample
12. The lengths of the sides of a triangular
garden are 18 feet, 14 feet, and 10 feet.
What is the measure of the largest angle
of the garden? Round to the nearest
tenth. 95.7º
sin 32° sin 40°
=
x
18
18 sin 40°
x=
sin 32°
x ≈ 21.8 yd
Lessons 10.7, 10.8, and 10.9
Review Examples
Find the exact value of cos 105°.
Student Review Exercises
Find the exact value of each expression.
Write 105° as 45° + 60° and use the angle sum
identity cos(A + B) = cos A cos B – sin A sin B.
13. tan 105°
− 3 −2
14. sin 75°
2+ 6
4
cos 105° = cos 45° • cos 60° − sin 45° • sin 60°
= 2 •1− 2
2 2 2
= 2− 6
4
4
= 2− 6
4
•
3
2
6− 2
4
1
300°
2
15. sin 15°
16. cos
Solve each equation.
17.tan2 θ – tan θ = 0, 0 ≤ θ < 2π
0, ≠ , ≠, 5≠
4
4
18.sin2 θ – 1 = 0, 0 ≤ θ < 360° 90°, 270°
19.sin2 θ – 1 = cos2 θ, 0 ≤ θ < 2π
≠ , 3≠
2 2
Chapter Review
503
Chapter Test
Solve each problem.
1. Solve for x. Round to the nearest tenth.
53.6 m
Use the Law of Sines or the Law of
Cosines to find each measure. Round to
the nearest tenth.
14.Find b if m∠B = 52º, m∠C = 31º,
and c = 12. 18.4
15. Find m∠A if a = 13.5, b = 9.4,
and c = 16.3. 55.9º
Solve each equation in the given interval.
2. Convert 70° to radians.
7≠
18
3.Convert 3≠ radians to degrees. 54°
10
Graph each function over the specified
domain. see margin
16.sin2 θ – 2sin θ – 3 = 0, 0 ≤ θ < 360°
270°
17.2sin θ + 1 = csc θ, 0 ≤ θ < 2π
≠ , 5≠ , 3≠
6 6 2
18. Find the area of triangle ABC below.
Round to the nearest whole number.
38 cm2
4. y = 3sin θ, 0 ≤ θ ≤ 2π
5. y = –1.5cos 2θ, –2π ≤ θ ≤ 2π
Find each function value. Round to the
nearest tenth.
6. g(θ) = 10 – 4sin θ, θ = 35º 7.7
7. y = –2.5tan x , x = 2≠ –4.3
2
3
Solve for x. Round to the nearest tenth
degree or thousandth radian.
8.sin x = 0.96, –90º ≤ x ≤ 90º 73.7º
9.cos x = 0.809, 0 ≤ x ≤ π 0.628 radians
Find the measure, in degrees, of each
angle x. Round to the nearest tenth if
necessary.
10.tan x = –1.1, –90º < x < 90º–47.7º
11.sin x = –0.965, –90º ≤ x ≤ 90º–74.8º
Use an angle identity to rewrite
each expression.
12.cos 35° cos 75° – sin 35° sin 75° cos 110°
19. An airliner is cruising at an elevation
of 6 miles. To make a smooth descent
for the passengers, the pilot begins his
descent at a distance of 70 miles from
the airport. What is the plane’s angle of
descent (the angle formed by the runway
and the flight path)? Round to the
nearest tenth. 4.9°
20. The voltage, V, in the power cord of a
computer t seconds after it was turned
on can be modeled using the function
V(t) = 120cos (120πt). After how many
seconds is the voltage equal to its
maximum? Express the answer so that
it represents all solutions in the interval
t > 0. n , where n is a whole number
13.sin 80° cos 40° – cos 80° sin 40° sin 40°
504 Chapter 10 Trigonometric Functions and Identities
60
Standardized Test Practice
Multiple Choice
Gridded Response
1. What is the value of cos–1 0.788? A
6.How many 4-inch cubes can be placed
inside a box that is 16 inches tall,
8 inches wide, and 12 inches long? 24
A38°
C49°
B42°
D56°
2. What is the common difference of the
arithmetic sequence below? C
18.4, 16.1, 13.8, 11.5, 9.2, …
A2.8
C–2.3
7.What is the value of x in the triangle
below? Round to the nearest tenth
of a centimeter. 22.1
B2.3
D–2.8
3. What is the slope of line m below? B
Open Ended Response
8.Describe how to convert an angle
measure of 20º from degrees to
radians. Then give the number of
radians. Multiply 20º by ≠ ; ≠ radians
180 9
B– 2
A– 3
2
3
2
3
D
C
3
2
4. Which angle measure is equivalent to
2≠ radians? D
5
A48°
B64°
C70°
D72°
5. Evaluate the trigonometric function
y = 6 – cos 2θ for θ = 20º. Round to the
nearest tenth. B
A5.1
B5.2
C5.3
D5.4
Extended Response
9. Two airplanes take off from the same
airport at the same time. Plane A flies in
a direction 24º east of north and cruises
at a speed of 550 miles per hour. Plane B
flies in a direction 38º west of north and
cruises at a speed of 490 miles per hour.
a. How far has each plane traveled after
90 minutes? Plane A: 825 mi; Plane B: 735 mi
b. Sketch a diagram to show the
position of each plane after
90 minutes. see margin
c. Find the distance between the
two planes after 90 minutes. Round
to the nearest tenth. 807.2 mi
Chapter Assessments
505