the-eye.eu Books/Mathemati… · 655 CHAPTER 6 Introduction to the Trigonometric Functions ACTIVITY 6.1 The Leaning Tower of Pisa OBJECTIVES 1. Identify the sides and corresponding

655

CHAPTER 6

Introduction to theTrigonometric Functions

ACTIVITY 6.1

The Leaning Tower of Pisa

OBJECTIVES

1. Identify the sides andcorresponding angles of aright triangle.

2. Determine the length of the sides of similarright triangles usingproportions.

3. Determine the sine,cosine, and tangent of an angle using a righttriangle.

4. Determine the sine,cosine, and tangent of anacute angle by using thegraphing calculator.

CLUSTER 1 Introducing the Sine, Cosine, and Tangent Functions

During a trip to Italy, you visit a wonder of the world, the Leaning Tower of Pisa.Your guidebook explains that the tower now makes an 85-degree angle with theground and measures 179 feet in height. If you drop a stone straight down fromthe top of the tower, how far from the base will it land?

To answer this question accurately, you need a branch of mathematics calledtrigonometry. Although the development of trigonometry is generally credited tothe ancient Greeks, there is evidence that the ancient Egyptian cultures usedtrigonometry in constructing the pyramids.

Right TrianglesAs you will see, the accurate answer to the Leaning Tower question requires someknowledge of right triangles. Consider the following right triangle with angles A,B, and C and sides a, b, and c.

A C

ac

b

B

DEFINITION

Angle C is the right angle, the angle measuring 90°. The side opposite the rightangle, c, is called the hypotenuse. Side a is said to be opposite angle A because itis not part of angle A. Side b is said to be adjacent to angle A because it and thehypotenuse form angle A. Similarly, side b is the side opposite angle B, and side ais the side adjacent to angle B.

Note that in any right triangle, the lengths of the sides are related by thePythagorean theorem.

In words, the square of the hypotenuse is equal to the sum of the squares of theother two sides.

c2 � a2 � b2

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Example 1 Consider the following right triangle.

The side opposite angle B is 12 centimeters long. The side opposite angle A is 5centimeters long. The side adjacent to angle B is 5 centimeters long. The side ad-jacent to angle A is 12 centimeters long. The hypotenuse is 13 centimeters long.Note so the Pythagorean theorem is satisfied.

1. Consider the following right triangle.

Determine the length of each of the following.

C A

1.5 in.1.2 in.

0.9 in.

B

132 � 52 � 122,

C A12 cm

13 cm5 cm

B

656 Chapter 6 Introduction to the Trigonometric Functions

a. the side opposite angle B

0.9 in.

b. the side adjacent to angle B

1.2 in.

c. the side opposite angle A

1.2 in.

d. the side adjacent to angle A

0.9 in.

e. the hypotenuse

1.5 in.

f. Demonstrate that the length ofthe sides satisfy the Pythagoreantheorem.

2.25 � 2.25

2.25 � 1.44 � 0.81

1.52 � 1.22 � 0.92

AAA

Similar TrianglesConsider the following right triangles whose angles are the same but whose sidesare different lengths. These three triangles are called similar triangles.

06-100 ch 06 pp3 1/6/07 10:43 AM Page 656

Activity 6.1 The Leaning Tower of Pisa 657

2. a. Using a protractor, estimate the measure of angle A to the nearest degree.(For a review of angles measured in degrees, see Appendix B.) Because

, angle A is called an acute angle.

approximately 38°

b. Use a metric or English ruler to complete the following table.

0° � A 6 90°

3. a. Use the information in the table in Problem 2b to complete the ratios in thefollowing table with respect to angle A. Write each ratio as a decimalrounded to the nearest tenth.

b. What do you observe about the ratio

for each of the three right triangles?

The ratio is the same for each of the three right triangles; that is, tothe nearest tenth, they are all 0.6.

length of the side opposite angle Alength of the hypotenuse

The table in Problem 3a illustrates the geometric principle that correspondingsides of similar triangles are proportional. Recall that the ratios and areproportional if ab �

cd.

cd

ab

4. Consider another right triangle in which the measure ofangle A is not the same as the measure of angle A in thethree similar triangles from Problems 2 and 3.

a. Using a protractor, estimate the measure of angle Ato the nearest degree.

Angle A to the nearest degree is 45°.

SMALL TRIANGLE 1.7 in. or 4.3 cm 1 in. or 2.5 cm 1.3 in. or 3.3 cm

MIDSIZE TRIANGLE 2.1 in. or 5.4 cm 1.3 in. or 3.4 cm 1.7 in. or 4.3 cm

LARGE TRIANGLE 2.6 in. or 6.8 cm 1.6 in. or 4.2 cm 2.1 in. or 5.4 cm

LENGTH OF THE LENGTH OF THE SIDE LENGTH OF THE SIDEHYPOTENUSE OPPOSITE A ADJACENT TO A

SMALL TRIANGLE 0.6 0.8 0.8

MIDSIZE TRIANGLE 0.6 0.8 0.8

LARGE TRIANGLE 0.6 0.8 0.8

LENGTH OF THE SIDE LENGTH OF THE SIDE LENGTH OF THE SIDEOPPOSITE ANGLE A ADJACENT TO ANGLE A OPPOSITE ANGLE A

LENGTH OF THE LENGTH OF THE LENGTH OF THE SIDEHYPOTENUSE HYPOTENUSE ADJACENT TO ANGLE A

A

BAppendix

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b. For this new triangle, use your ruler to complete the following table withrespect to angle A.

c. Are the ratios for the new triangle the same as the ratios for the three sim-ilar triangles?

The ratios here are different from the ones in the table for the similartriangles.

d. What changed from the similar triangles to the new triangle to make theratios change?

The measure of angle A changed from 38° to 45°.

Sine, Cosine, and Tangent FunctionsAs indicated in Problem 4, the ratios of the sides of a right triangle are dependenton the size of the angle A. If the angle changes, the ratios change. This fact is fun-damental to trigonometry.

The ratios of the sides of a right triangle with acute angle A are a function of thesize of A.

The ratios are given special names and are defined as follows.

DEFINITION

Let A be an acute angle (less than 90°) of a right triangle. The sine, cosine, andtangent of angle A are defined by

where sin, cos, and tan are the standard abbreviations for sine, cosine, and tan-gent, respectively.

• Sine, cosine, and tangent are called trigonometric functions.

• Note that the input values for the trigonometric functions are angles and the out-put values are ratios.

• A mnemonic device often used to help remember the trigonometric relationshipsis SOH CAH TOA, where SOH indicates the sine function is the length of theside opposite over the length of the hypotenuse, CAH indicates the cosine defini-tion and TOA indicates the definition of the tangent function.

tangent of A � tan A �length of the side opposite A

length of the side adjacent to A,

cosine of A � cos A �length of the side adjacent to A

length of the hypotenuse,

sine of A � sin A �length of the side opposite A

length of the hypotenuse,

NEW TRIANGLE 0.7 0.7 1.0

LENGTH OF THE SIDE LENGTH OF THE SIDE LENGTH OF THE SIDEOPPOSITE ANGLE A ADJACENT TO ANGLE A OPPOSITE ANGLE A

LENGTH OF THE LENGTH OF THE LENGTH OF THE SIDEHYPOTENUSE HYPOTENUSE ADJACENT TO ANGLE A

06-100 ch 06 pp3 1/6/07 10:43 AM Page 658

Example 2 Consider the following right triangle with the given approximatedimensions.

or 0.6 (nearest tenth)



Therefore, for any size right triangle with a 38° angle as one of its acute angles,the sine of 38° is always approximately 0.615, the cosine of 38° is always approx-imately 0.807, and the tangent of 38° is always approximately 0.761.

tan 38° �1.62.1

� 0.761

cos 38° �2.12.6

� 0.807

sin 38° �1.62.6

� 0.615

2.1

2.6

38°

1.6


You can use your graphing calculator to evaluate sin 38°. Make sure the calculatoris in degree mode. Press the key, followed by , , and .

5. a. The sum of the three angles in a right triangleis 180°. If one of the acute angles measures 38°, the other acute angle is 52°. Use the accompanying figure to determine each of the following.

i. 0.808 or 0.8 (nearest tenth)

ii. or 0.6 (nearest tenth)

iii. or 1.3 (nearest tenth)

b. Verify your rounded answers in part a using your graphing calculator.

�2.11.6

� 1.313tan 52°

�1.62.6

� 0.615cos 52°

�2.12.6

�sin 52°

ENTER)83SIN

2.1

2.6

38°

52°

1.6

06-100 ch 06 pp3 1/6/07 10:43 AM Page 659

Example 3 Consider the following right triangle where A and B are acuteangles.

, , , , ,


Calculate the following (4 decimal places).

a. 0.8000 b. 0.6000 c. 1.3333

d. 0.6000 e. 0.8000 f. 0.7500

Example 4 Solve the following equations. Round your answers to the nearesttenth.

a. b. c.

SOLUTION

Using the calculator in degree mode to evaluate the function values,

tan 72° �x

24 .7cos 13° �24xsin 56° �

x15

tan Bcos Bsin B

tan Acos Asin A

C A

1.5 in.1.2 in.

0.9 in.

B

tan B �125

cos B �513

sin B �1213

tan A �512

cos A �1213

sin A �513

C A12 cm

13 cm5 cm

B


a. (0.829) �1510.8292 � x

12.4 � x

x15

7. Solve the following equations. Round your answers to the nearest tenth.

a.x � 4.1

sin 24° �x

10

b. 0.974 �10.9742x � 24

x � 24.6

x �24

0.974

24x

c. 3.098 �13.0782 124.72� x

76.0 � x

x24.7

b.x � 10.7

cos 63° �x

23.5 c.x � 14.4

tan 48° �16x

06-100 ch 06 pp3 1/6/07 10:43 AM Page 660

Tower of Pisa Problem RevisitedYou are now ready to answer the original Tower of Pisa problem.

8. a. Recall that the tower makes an 85° angle with the ground and the tower is179 feet tall. Construct a right triangle that satisfies these conditions.

b. With respect to the 85° angle, is the height of the tower represented by thelength of an opposite side, the length of an adjacent side, or the length ofthe hypotenuse of your triangle?

The length of the tower is represented by the hypotenuse with re-spect to the 85° angle. The height of the tower (distance from top oftower to base measured vertically) is measured by the length of theopposite side.

c. You want to determine how far from the base of the tower the stone hitsthe ground. Therefore, you want to determine the length of which side ofthe triangle with respect to the 85° angle?

I want to determine the side adjacent to the 85° angle.

d. Which trigonometric function relates the side with the length you knowand the side with the length you want to know?

The cosine function relates the 179-foot side (hypotenuse) to the dis-tance the tower leans from the vertical (adjacent).

e. Write an equation using the information in parts a–d.

where x is the side adjacent to the 85° angle in the dia-gram in part a.

f. Using your calculator to evaluate solve the equation in part e.

ft.

Thus, the tower leans 15.6 ft from the vertical.

x � 15.6

cos 85°,

cos 85° �x

179,

A C

B

Tower179 ft

y

x

85°


06-100 ch 06 pp3 1/6/07 10:43 AM Page 661


1. The trigonometric functions are functions whose inputs are measures ofthe acute angles of a right triangle and whose outputs are ratios of thelengths of the sides of the right triangle.

2. The three sides of a right triangle are the adjacent side, the opposite side,and the hypotenuse. The hypotenuse is always the side opposite the right190°2 angle. The other two sides vary, depending on which angle is used asthe input.

3. The sine, cosine, and tangent of the acute angle A of a right triangle aredefined by

For additional practice working with the trigonometric functions in righttriangles and special triangles, see Appendix B.

tan A �length of the side opposite A

length of the side adjacent to A.

cos A �length of the side adjacent to A

length of the hypotenuse

sin A �length of the side opposite A

length of the hypotenuse

SUMMARYACTIVITY 6.1

EXERCISESACTIVITY 6.1

1. Triangle ABC is a right triangle.

Determine each of the following.

C

B

A

35

4

Exercise numbers appearing in color are answered in the Selected Answers appendix.

a.0.6000

b.0.8000

c.0.8000

cos Asin Bsin A

d.0.6000

e.0.7500

f.1.3333

tan Btan Acos B

BAppendix

06-100 ch 06 pp3 1/6/07 10:43 AM Page 662

2. In a certain right triangle,

a. Determine possible lengths of the three sides of aright triangle.

Hint: Use the Pythagorean theorem,

to determine the length of any unknown side.

If the length of the side opposite angle A is 24;the hypotenuse is 25. Using the Pythagoreantheorem, I determine that the length of theside adjacent to angle A is 7.

b. Determine .

c. Determine .247 � 3.4286

tan A

725 � 0.2800

cos A

c2 � a2 � b2,

sin A �2425.


2524

7A

3. In a certain right triangle,

a. Determine possible lengths of the three sides of the right triangle.

Given the length of the side opposite angle B is 7; thelength of the side adjacent to angle B is 4. Using the Pythagoreantheorem, I determine that the length of the hypotenuse is

b. Determine

c. Determine

4. Consider the accompanying right triangle.

a. Which of the trigonometric functions relatesangle A and sides a and x?

the tangent function

b. What equation involving angle A and side awould you solve to determine the value of x?

tan A �ax

cos B �4265

� 0.4961

cos B.

sin B �7265

� 0.8682

sin B.

265.

tan B �74 ,

tan B �74.

Ax

a

5. Given angle B and side c in the accompanying diagram, an-swer the questions in parts a and b.

a. Which of the trigonometric functions relates angle B andsides c and y?

the sine function

b. What equation involving angle B and side c would yousolve to determine the value of y?

sin B �yc

C

B

y

c

06-100 ch 06 pp3 1/6/07 10:43 AM Page 663

6. Consider the accompanying right triangle.

a. Which of the trigonometric functions relates angle A and sides b and z?

the cosine function

b. What equation involving angle A and side b would you solve to determinethe value of z?

cos A �bz

A Bz

b


7. Solve the following equations. Round your answer to the nearest tenth.

a. b. c.

x � 61.4

x �23

sin 22°

x sin 22° � 23

sin 22° �23x

x � 85.6

x � 9 tan 84°

tan 84° �x9

x � 9.1

12 sin 49° � x

sin 49° �x

12

8. A friend asks you to help build a ramp at his mother-in-law’s house. Thereare three 7-inch-high steps leading to the front door. Another friend donatesa 15-foot ramp. The building inspector informs you that any handicappedramp can have an inclination no greater than 5°.

a. Sketch a diagram that assumes the land in front of the steps is level andthat the ramp makes a 5° angle with the top of the steps.

b. What is the increase in height from one end of the ramp to the other?

in. The increase in height from one end of the ramp to thetop of the stairs is ft.

c. Would the donated ramp be long enough to meet the code? Explain.

ft.

The ramp needs to be at least 20.1 feet long. Therefore, the donatedramp will not be long enough to meet the code.

Alternative approach: ft. The three steps must measureat most 1.3-feet high for the 15-foot ramp to satisfy the code.

Each solution suggests ways to think about modifications to eitherthe ramp or the steps (or both) that could be used to meet the code.

15 sin 5 � 1.3

x � 20.1

x �1.75sin 5°

sin 5° �1.75

x

2112 � 1.75

3 # 7 � 21

3 steps: Total height = 1.75 ft.Ramp = x ft

5°

5°

Note: Not to scale

06-100 ch 06 pp3 1/6/07 10:43 AM Page 664

You and a friend take a camping trip to the American Southwest. On the wayhome, you realize that your Jeep is running low on gas, so you stop at a filling sta-tion. Unfortunately, the attendant informs you that his station has been withoutgas for several days. However, he is certain that there is a station on a side road upahead that should have gas to sell.

A quick phone call confirms this information, and you begin to ask about the exactlocation of the station. The attendant’s map of the area indicates that the station isabout 10 miles away at an angle of 35° with the road you are currently traveling.

Your Jeep is equipped with a compass, so driving through the desert is not a greatproblem, although you would prefer to stay on the road. You estimate that youhave enough gas for 15 miles. Do you have enough gas to make it to the other sta-tion without going through the desert?

This problem can be solved using the triangle above and some trigonometry to de-termine the distance from you to the intersection, side YI, and from the intersec-tion to the gas station, side IG.

1. With respect to angle Y, select the correct response in parts a and b.

a. the side YI is known as the

i. side opposite angle Y,

ii. the side adjacent to angle Y, or

iii. the hypotenuse

b. the side YG is known as



iii. the hypotenuse

c. the function that relates YI, YG, and angle Y is the .

2. a. Set up an equation indicated by Problem 1c.

b. Solve the equation for the length of YI.

mi. YI � 8.2 YI � 10 cos 35

cos 35° �Y I10

cos Y �Y IYG

cosine function

Main road(Your location)

Y

Side road

35°

10 mi.

I (Intersection)

G (Gas station)

Desert

Activity 6.2 A Gasoline Problem 665

ACTIVITY 6.2

A Gasoline Problem

OBJECTIVES

1. Identify complementaryangles.

2. Demonstrate that the sineof one of the complemen-tary angles equal thecosine of the other.

06-100 ch 06 pp3 1/6/07 10:43 AM Page 665

3. With respect to angle Y, select the correct response in parts a and b.

a. the side IG is known as



iii. the hypotenuse




iii. the hypotenuse

c. the function that relates IG, YG, and angle Y is the .


b. Solve the equation for the length of IG.

mi.

5. Do you have enough gas to make it without journeying through the desert?Explain.

To avoid the desert, I would have to travel miles on thetwo roads to the other station. I should be able to make it.

Meanwhile at the second service station, the attendant is also making some calcu-lations. He is aware of your situation and is trying to anticipate which option youwill choose in case he has to go look for you.

He knows that the line to the first station makes a 55° angle with his road.

6. With respect to angle G, select the correct response in parts a and b.

a. the side YI is known as the

i. side opposite angle G,

ii. the side adjacent to angle G, or

iii. the hypotenuse

Main road(Your location)

Y

Side road

35°

55°

10 mi.

(I) Intersection

G (Second gas station)

Desert

8.2 � 5.7 � 13.9

IG � 5.7

IG � 10 sin 35°

sin 35° �IG10

sin Y �IGYG

sine function


06-100 ch 06 pp3 1/6/07 10:43 AM Page 666




iii. the hypotenuse

c. the function that relates YI, YG, and angle G is the .


b. Solve the equation for the length of YI.

mi.

c. How does the answer in part b compare to the answer in Problem 2b?

They are the same.

8. With respect to angle G, select the correct response in parts a and b.

a. the side IG is known as



iii. the hypotenuse




iii. the hypotenuse

c. the function that relates IG, YG, and angle G is the .


b. Solve the equation for the length of IG.

mi.

c. How does the answer in part b compare to the answer in Problem 4b?

They are the same.

Complementary AnglesAlthough you used different angles of the triangle YGI in Problems 1–9, your re-sults for the lengths of YI and IG should have been the same. The angles involvedin these calculations were 35° and 55°. Because their sum is 90°, they are calledcomplementary angles.

IG � 5.7

IG � 10 cos 55°

cos G �IGYG

cosine function

YI � 8.2

YI � 10 sin 55°

sin 55° �YI10

sin G �YIYG

sine function


DEFINITION

Two acute angles A and B whose measures sum to 90° are called complementaryangles. If x represents the measure of an acute angle, then represents themeasure of the complementary angle.

90 � x

06-100 ch 06 pp3 1/6/07 10:43 AM Page 667

Example 1


10. a. The solution to Problem 2b involved and the solution to Problem7b involved Compare the values of and

in the respective equations; thus, the resulting valuesare equal.

b. The solution to Problem 4b involved and the solution to Problem9b involved Compare the values of and

in the respective equations; thus, the resulting valuesare equal.

This situation demonstrates another fundamental principle of trigonometry.

sin 35° � cos 55°

cos 55°. sin 35°cos 55°. sin 35°

cos 35° � sin 55°

sin 55°.cos 35° sin 55°.cos 35°

Cofunctions of complementary angles are equal. Symbolically, if x is an acuteangle, then

and

In fact, the name cosine is derived from the words sine and complement.

cos x � sin 190° � x 2.

sin x � cos 190° � x 2

Example 2

a. b.c. d.

11. Complete the following table, where x represents the measure of an angle indegrees. Use your graphing calculator to determine values of and

(4 decimal places).cos 190 � x 2sin x

sin 18° � cos 72°sin 72° � cos 18°sin 55° � cos 35°sin 35° � cos 55°

35 55

55 35

72 18

18 72

x, 90 � x,ACUTE ANGLE COMPLEMENTARY ANGLE

0° 90° 0.0000 0.0000

15° 75° 0.2588 0.2588

30° 60° 0.5000 0.5000

45° 45° 0.7071 0.7071

60° 30° 0.8660 0.8660

75° 15° 0.9659 0.9659

90° 0° 1.0000 1.0000

x 1190 � x22° sin x cos 1190 � x22°

06-100 ch 06 pp3 1/6/07 10:43 AM Page 668


1. Complementary angles are two acute angles, whose measures sum to 90°.

2. The co in cosine is from the word complement.

3. Cofunctions of complementary angles are equal.

SUMMARYACTIVITY 6.2


1. You are in a rowboat on Devil Lake in Ontario, Canada. Your lakeside cabin hasno running water, so you sometimes go to a fresh spring at a different point onthe lakeshore. You row in a direction 60° north of east, for half a mile. You arenot yet tired, and it is a lovely day.

a. How far would you now have to row if you were to return to your cabin byrowing directly south and then directly west? Round your answer to thenearest hundredth.

mi.

mi.

mi.

I would travel 0.68 miles going back directly south and then directly west.

b. Check your solution to part a, using the complementary angle and the co-function of those used in part a.

mi.

mi.

These calculations confirm the result in part a.

2. One afternoon you don’t pay enough attention while rowing. You row 20° offcourse, too far north, instead of directly west as you had intended. You row offcourse for 300 meters.

a. Draw a diagram of this situation.

w � 0.5 sin 30° � 0.25

s � 0.5 cos 30° � 0.43

s � w � 0.68

w � 0.5 cos 60° � 0.25

s � 0.5 sin 60° � 0.43

60°

South

w, distancerowed west

s, distance rowed south

0.5mi.

North

EastWest


20°

300 m

x

y b. How far west have you gone? Round your answer to the nearest meter.

Let x be the distance to the west when I realize I am off course.

m

I have rowed 282 meters directly west.

x � 300 cos 20° � 282

06-100 ch 06 pp3 1/6/07 10:43 AM Page 669

c. How far north are you of where you had originally planned to be?

Let y be the distance I have rowed to the north when I realize I amoff course.

m

I am 103 meters to the north when I realize I am off course.

d. Check your solutions to parts b and c, using the complementary angle andthe cofunctions of those used in parts b and c.

m

m

These calculations verify the results in b and c.

y � 300 cos 190°�20° 2 � 300 cos 170° 2 � 103

x � 300 sin 190°�20° 2 � 300 sin 170° 2 � 282

y � 300 sin 20° � 103


3. a. Complete the following table, where x represents the measure of an anglein degrees. Use your graphing calculator to determine values of and

to 4 decimal places.cos 190 � x 2sin x

7° 83° 0.1219 0.1219

17° 73° 0.2924 0.2924

24° 66° 0.4067 0.4067

33° 57° 0.5446 0.5446

48° 42° 0.7431 0.7431

67° 23° 0.9205 0.9205

77° 13° 0.9744 0.9744

x 1190 � x22° sin x cos 1190 � x22°

b. What trigonometric property does this table illustrate?

The table in part a illustrates the property that cofunctions of com-plementary angles are equal.

06-100 ch 06 pp3 1/6/07 10:43 AM Page 670

Activity 6.3 The Sidewalks of New York 671

ACTIVITY 6.3

The Sidewalks of New York

OBJECTIVES

1. Determine the inversetangent of a number.

2. Determine the inversesine and cosine of anumber using thegraphing calculator.

3. Identify the domain andrange of the inverse sine,cosine, and tangentfunctions.

A friend of yours is having a party in her Manhattan apartment, which bordersCentral Park. She gives you the following directions from your place, which alsoborders the park.

i. If you are coming after dark, head east for three blocks, going around thepark, and then go north for two blocks.

ii. If you can come early, you can cut through the park, a shorter route.

You leave your place early and decide to cut through the park. You want tocompute the shortest distance, D (in blocks), between your apartments.

1. Using the Pythagorean theorem, determine the distance, D, in

blocks, across the park.

blocks

The shortest distance between the two apartments is 3.6 blocks.

Now you want to determine the direction (angle) you need to go to get to yourfriend’s place. You can represent the angle by u, the Greek letter theta. Becauseyou now have the lengths of all three sides of the triangle, you can use any of thetrigonometric functions to help determine u. Begin with the tangent function.

Inverse Tangent FunctionRecall that the input for the tangent function is an angle and the output is the ratio ofthe length of the side opposite to the length of the side adjacent in a right triangle.

2. Determine the value of the tangent of u, written tan u, for the Central Parktriangle.

If you want to determine the tangent of a known angle, you can use the keyon your calculator. From Problem 2, you know the value of the tangent of theangle u, but do not know the value of the u. That is, you know the output for thetangent function, but you do not know the input.

3. Use the table feature on your calculator to approximate u from Problem 2.You should compute the tangent of several possible values of u to get as close

TAN

tan u �23 � 0.6667

tangent of an angle �length of the opposite sidelength of the adjacent side

222 � 32 � 213 � 3.6

c2 � a2 � b2,

3 blocksYour place

2 blocks

θ

Her place

D

Central Park

06-100 ch 06 pp3 1/6/07 10:43 AM Page 671

Just as logx � y has the equivalent exponential form 10y � x, tan�1x � u isequivalent to tanu � x.

Example 1 Consider the following triangle.

or

Using the calculator, you can determine that See the followingTI-83/TI-84 Plus screen. Be careful, your calculator must be in degree mode.

The inverse tangent function is located on your calculator as second function totangent. That is, you will need to press the key before you press the tangentkey. Note that your calculator uses the notation rather than arctan.tan

�12nd

u � 39.8°.

u � tan �1a

56btan u �

56

6 m

5 m

θ


DEFINITION

The input of the inverse tangent function is the ratio

for the angle. The output is the acute angle. The inverse tangent function is denoted by tan�1 or arctan and defined by

tan�1 x � u, where u is the acute angle whose tangent is x.

Thus the input, x, represents the ratio of the length of the opposite side to thelength of the adjacent side for the angle u.

length of the opposite sidelength of the adjacent side

uu tan uu

30° 0.57735

40° 0.83910

35° 0.70021

34° 0.67451

33.7° 0.66692

33.69° 0.66666

You don’t have to experiment every time to determine u when you know thetangent of u. There is a more direct method using the inverse tangent function.Recall that with inverse functions, the inputs and outputs are interchanged.

as possible to the desired answer. Complete the following table. Round youranswers to five decimal places.

06-100 ch 06 pp3 1/6/07 10:43 AM Page 672

Using either the inverse sine or the inverse cosine function, you determine thatSee the following TI-83/TI-84 Plus screen.u � 36.9°.

4 ft

5 ft3 ft

θ

Again, sin�1x � u has the equivalent form sinu � x and cos�1x � u has theequivalent form cosu � x.

Example 2 Consider the following triangle.


DEFINITION

The input of the inverse sine function is the ratio

for the angle. The output is the acute angle. The inverse sine function is denotedby sin�1 or arcsin and defined by

sin�1x � u, where u is the acute angle whose sine is x.

Thus the input, x, represents the ratio of the length of the opposite side to thelength of the hypotenuse for the angle u.

The input of the inverse cosine function is the ratio

for the angle. The output is the acute angle. The inverse cosine function is denotedby cos�1 or arccos and defined by

cos�1x � u, where u is the acute angle whose cosine is x.

Thus the input, x, represents the ratio of the length of the adjacent side to thelength of the hypotenuse for the angle u.

length of the adjacent sidelength of the hypotenuse

length of the opposite sidelength of the hypotenuse

4. Use your calculator to determine the inverse tangent of the answer to Prob-lem 2. (Make sure your calculator is in degree mode.) Compare this answerwith your approximation from Problem 3.

Remember:

ratio

The value is very close to the approximation.

Inverse Sine and Cosine FunctionThere are similar definitions for the inverse sine and inverse cosine functions.

tan �110.6667 2 � 33.7°

tan 1angle 2 �

tan �11ratio 2 � angle

, so

, so u � cos �11452cos u �

45

u � sin �11352sin u �

35

06-100 ch 06 pp3 1/6/07 10:43 AM Page 673

5. a. In Problems 2 and 4 you determined the values of in the Central Parksituation using the inverse tangent function. Now use the inverse sinefunction to determine the value of in the Central Park situation.

b. Use the inverse cosine function to determine the value of u in the CentralPark situation.

6. The term used by highway departments when describing the steepness of ahill is percent grade. For example, a hill with a 5% grade possesses a slopeof or This means there will be a 5-foot vertical change for every 100feet of horizontal change or 1 foot of vertical change for every 20 feet of hor-izontal change as a car ascends or descends the hill.

a. You are driving along Route 17 in the Catskill Mountains of New YorkState. Just before coming to the top of a hill, you spot a sign that reads“7% Grade Next 3 Miles Trucks Use Lower Gear.” Draw a triangle, andlabel the appropriate parts to model this situation.

b. Use an inverse trigonometric function to determine the angle that the baseof the hill makes with the horizontal.

c. Use trigonometry to determine how many feet of elevation you will losefrom the top of the hill to the bottom.

mi.

In 3 miles, the elevation drops 0.21 miles or 1106 feet.

y � 3 sin 14.004 2 � 0.21

u � tan�11 7100 2 � 4.004°

y

θ

3 mi.

120.5

100

u � cos�1 1 32132 � 33.6901°

u � sin�1 1 22132 � 33.6901°

u

u


1. The domain (set of inputs) of each inverse trigonometric function is a setof ratios of the lengths of the sides of a right triangle.

2. The range (set of outputs) of each inverse trigonometric functions is the setof angles.

3. For inverse trigonometric functions

is equivalent to

is equivalent to and

is equivalent to tan u � x.tan �1

x � u

cos u � x,cos �1 x � u

sin u � x,sin �1

x � u

SUMMARYACTIVITY 6.3

06-100 ch 06 pp3 1/6/07 10:43 AM Page 674

2. Complete the accompanying table, which refers to right triangles, labeled as inthe figure below.

V

θ

N

E


1. For each of the following, use your calculator to determine u to the nearest0.01°.


a. b.u � 64.62°

u � cos �11372

u � 30°

u � arcsin 1122

c. d.u � 63.82°

u � sin �1 10.8974 2

u � 67.04°

u � arctan 12.36 2

e. f.u � 64.62°

cos u �37

u � 66.80°

tan u �73

g. h.u � 20.76°

tan u � 0.3791

u � 22.28°

sin u � 0.3791

V uu E N

32 65° 13.5 29.0

23.3 59° 12 20

4.1 54° 2.4 3.3

26 43.8° 18.8 18

4.5 45° 3.2 3.2

3. While hiking, you see an interesting rock formation on the side of a verticalcliff. You want to describe for a friend how he might see it when he walks downthe path. If you stand on the path at a certain place, 50 feet from the base of thecliff, the rock formation is visible about 30 feet up the cliff. At what angleshould you tell your friend to look?

My friend should look up at an angle of approximately 31°.

u � tan �1 130

502 � 31°

30 ft.

θ50 ft.


06-100 ch 06 pp3 1/6/07 10:43 AM Page 675

4. A warehouse access ramp claims to have a 10% grade. The ramp is 15 feetlong.


b. What angle does the ramp make with the horizontal?

Because grade is rise over run,

The ramp makes an angle of 5.7° with the horizontal.

c. How much does the elevation change from one end of the ramp to theother?

ft.

The elevation changes 1.5 feet from one end of the ramp to the other.

y � 15 sin 15.7 2 � 1.5

u � tan �1

10.1 2 � 5.7°

tan u � 0.1.

15 ft.

NOT TO SCALE

θ


06-100 ch 06 pp3 1/6/07 10:43 AM Page 676

Activity 6.4 Solving a Murder 677

There has been a fatal shooting 110 feet from the base of a 25-story building. Eachstory measures approximately 12 feet. Two suspects live in the building, one onthe 7th floor, the other on the 20th. Both suspects were in their apartments at thetime of the murder. Forensic specialists report that the bullet was fired from some-where in the building and entered the body at an angle of approximately 58° withthe ground.

1. Draw a diagram of this situation.

Based on the diagram, the question is now: Is or is equal to 58°?

2. Use the appropriate trigonometric function to determine which of the twosuspects could not have committed the murder.

These calculations show that the suspect on the 7th floor probably couldnot have committed the murder.

3. On what other floors of the building should the police question additionalpossible suspects? Explain.

ft. or about 14.7 floors

The police should question possible suspects on the 14th and 15th floors.

As demonstrated in Problems 2 and 3, determining the measure of the sidesand angles of a right triangle can be useful.

y � 110 tan 58° � 176

58°

110 ft

y

u � tan �1 1240

1102 � 65°

a � tan �1 1 84

1102 � 37°

ua

Base of building110 ft.θ

α

Shooting

84 ft.

240 ft.

ACTIVITY 6.4

Solving a Murder

OBJECTIVE

1. Determine the measure ofall sides and all angles ofa right triangle.

06-100 ch 06 pp3 1/6/07 10:43 AM Page 677

Example 1 Solve the right triangle ABC, with andinches.

SOLUTION

Consider the following diagram.

You need to determine the measurements of the remaining sides and angles; angleB, side a, and side b. Because and ,

To determine the length of side a, you can use angle A, side c, and the sine function.

So inches to the nearest tenth. See the followingcalculator screen.

To determine the length of side b, you can use angle A, side c, and the cosinefunction.

4. Use the given information to solve each of the following right triangles.

a. the side opposite the 43° angle: mi.

hypotenuse: mi.

the other acute angle: 90° � 43° � 47°

6cos 43° � 8.2

6 # tan 43° � 5.6

43°6 mi.

b � 12.2 cos 33° � 12.210.83872 � 10.2 in.

cos 33° �b

12.2

cos A �bc

a � 6.612.2 sin 33° � a,

sin 33° �a

12.2

sin A �ac

B � 90° � 33° � 57°.

33 � B � 90, or

A � 33°A � B � 90°

c = 12.2 in.

33°

a

A C

B

b

c � 12.2C � 90.0°,A � 33.0°,


DEFINITION

To solve a triangle means to determine the measure of all sides and all angles.This process can be especially useful for architects, surveyors, and navigators.

06-100 ch 06 pp3 1/6/07 10:43 AM Page 678


b. hypotenuse: by using thePythagorean theorem

the angle adjacent to side 8:

the angle adjacent to side 5: 58°

c. the other leg:

the angle adjacent to side 9:

the other acute angle: 27°

cos �1 1 9202 � 63°

2319 � 17.920

9

tan �1 1582 � 32°

289 � 9.48 5

1. Many trigonometric problems involve solving right triangles, that is,determining the measures of all sides and angles.

2. The following is a trigonometric problem-solving strategy:

a. Draw a diagram of the situation using right triangles.

b. Identify all known sides and angles.

c. Identify sides and/or angles you want to know.

d. Identify functions that relate the known and unknown.

e. Write and solve the appropriate trigonometric equation(s).

SUMMARYACTIVITY 6.4


1. Use the given information to solve each of the following right triangles.

a. The side adjacent to the 57° angle is 4.2 feet.

The hypotenuse is 7.8 feet.

The other acute angle is 33°.

57°

6.5 ft.


b. The hypotenuse is 19.0.

The angle adjacent to side 18 is 18.4°.

The other acute angle is 71.6°.

186

c. The other leg is 7.9 inches.

The angle adjacent to side 9 inches is 41.4°.

The other angle is 48.6°.

12 in.

9 in.

06-100 ch 06 pp3 1/6/07 10:43 AM Page 679

2. You need to construct new steps for your deck and read that stringers are onsale at the local lumber company. Stringers are precut side supports to whichyou nail the steps; they are made in three-, four-, five-, six-, or seven-stepsizes. Each step on the stringers is 7 inches high and 12 inches deep.

a. If the vertical rise of your deck measures 3.5 feet, which size stringershould you buy? Explain.

The height of the staircase is inches. The number ofsteps Therefore, I will need the 6-step stringer.

b. How far out will your steps extend from the porch? Explain.

The steps will extend inches, or 6 feet fromthe porch.

c. What angle will a line from the lower right end of the stringer to the edgeof the deck make with the ground? Explain.

The height of the staircase is 3.5 feet, and the length is 6 feet.

The line makes an angle of about 30° with the ground.

d. What angle will the line in part c make with the vertical? Explain.

59.7°, because the sum of the angles of a triangle is 180°.

3. Some application problems involve a horizontal line of sight, which is usedas a reference line. An angle measured above the sight line is called an angleof elevation. An angle measured below the sight line is called an angle ofdepression.

Sight lineAngle of elevation

Angle of depression

u � tan �1 13.5

6 2 � 30.3°

3.5 ft.

6 ft.θ

6 steps � 12 inches � 72

�42 in.7 in. � 6.

3.5 � 12 � 42

12 in.

7 in.


06-100 ch 06 pp3 1/6/07 10:43 AM Page 680


You and some friends take a trip to Colton Point State Park in the GrandCanyon of Pennsylvania. Some of your group go white-water rafting, whilesome of your friends join you for a hike. You reach the observation deck intime to see the rest of your party battling the white water. Someone in thegroup asks you how close you think the rafts actually get to you as they floatby. You have no idea, but you ask a nearby park ranger.

She doesn’t know either, but she does tell you that the canyon is approxi-mately 800 feet deep at Colton Point and that the angle of depression to thecreek is about 22°.


b. Use trigonometry to estimate how close the rafters get to you on theobservation deck.

The direct distance, d, from the observation deck to the raft is

approximately feet.

If you could walk straight down the cliff and straight across at thebase of the cliff to the creek, the distance would be approximately

feet, where .b �800

tan 22b � c � 2780

800sin122° 2 or 2135

c � 800 ft.

b

d: direct distance from observation deck to raft/creek

Observation deck

22°

22°Raft/creek

06-100 ch 06 pp3 1/6/07 10:43 AM Page 681

Situation 1: Stabilizing a TowerYou are considering buying property near a cell phone tower and are concernedabout your property values. You decide to do some reading about issues involvingtowers, such as aesthetics, safety, and stability. Of course, anyone living near thetower would like a guarantee that it could not blow down. Guy wires are part ofthat guarantee.

The tower rises 300 feet and is supported by several pairs of guy wires all attachedon the ground at the same distance from the base of the tower. In each pair, oneguy wire extends from the ground to the top of the tower, and the other attacheshalfway up. The accompanying diagram illustrates one pair of wires.

1. New guidelines for stability recommend that the angle (A) the guy wiresmake with the line through the center of the tower (not the angle ofelevation) must be at least 40°. Therefore, the existing guy wires may needto be replaced. Because you are concerned about how close the wires willcome to your property, you need to compute the shortest distance from thetower at which the guy wires may be attached. Use trigonometry tocompute this distance.

Let x represent the distance of the guy wire from the center of the baseof the tower.

ft.

Because the angle must be at least 40°, the minimum distance from thetower at which the guy wires may be attached is 252 feet.

2. To improve stability, the authors of the guidelines propose increasing theminimum angle the guy wire makes with the tower from 40° to 50°. What isthe effect on the shortest distance from the tower at which the guy wires maybe attached?

The shortest distance increases from about 252 feet to about 358 feet,because feet.

Situation 2: Climbing a MountainYou are camping in the Adirondacks and decide to climb a mountain that dominatesthe local area. You are curious about the vertical rise of the mountain and recall from

300 # tan 50° � 357.53

x � 300 # tan 40° � 252

300 ft.

x

A


PROJECTACTIVITY 6.5

How Stable Is ThatTower?

OBJECTIVEs

1. Solve problems usingright-triangletrigonometry.

2. Solve optimization prob-lems using right-triangletrigonometry with agraphing approach.

06-100 ch 06 pp3 1/6/07 10:43 AM Page 682

project Activity 6.5 How Stable Is That Tower? 683

your mathematics course that surveyors can measure the angle of elevation of amountain summit with a theodolite. You are able to borrow a theodolite from thelocal community college to gather some pertinent data. You find a level field andtake a first reading of an angle of elevation of 23°. Then you walk 100 feet towardsthe mountain summit and take a second reading of 24° angle of elevation (seeaccompanying illustration).

3. Use h to represent the vertical rise of the mountain, and write expressions fortan 23° and tan 24° in terms of h and x.

a. b.

4. a. Solve the system of equations in Problem 3 to determine x. Round youranswer to the nearest tenth.From 3a,

From 3b,

ft.

b. Use the value of x from part a to determine the vertical rise, h, of themountain.

The vertical rise of the mountain is about 911 feet. To check,

Situation 3: Seeing Abraham LincolnYou are traveling to South Dakota and plan to see Mount Rushmore. In prepara-tion for your trip, you do some research and discover that from the observationcenter, the vertical rise of the mountain is approximately 500 feet and the heightof Abraham Lincoln’s face is 60 feet (see accompanying diagram). To get the best

h � 2145.3 tan 23° � 910.6.

h � 2045.3 tan 24° � 910.6

x � 2045.3

x �100 tan 23°

tan 24° � tan 23°

x1tan 24°�tan 23° 2 � 100 tan 23°

x tan 24°�x tan 23° � 100 tan 23°

100 tan 23° � x tan 23° � x tan 24°

h � 1tan 24° 2 1x 2

h � 1tan 23° 2 1100 � x 2

tan 24° �hxtan 23° �

h100 � x

24°

h

23°100 ft. x

06-100 ch 06 pp3 1/6/07 10:43 AM Page 683

view, you want to position yourself so that your viewing angle of Lincoln’s face isas large as possible.

5. a. Determine from the diagram which angle is the viewing angle.

b. Write an appropriate trigonometric equation for the angle interms of x.

c. Write an appropriate trigonometric equation for in terms of x.

d. Write the equivalent inverse function expression for the equations in partsb and c.

e. Write an equation that defines the viewing angle as a function of the dis-tance, x, that you are standing from the base.

b � tan�1 a500

xb�tan�1 a

440xb

a � tan�1 a440

xb

a � b � tan�1 a500

xb

tan a �440

x

a

tan 1a � b 2 �500

x

1a � b 2

b

Px

500 ft.

60 ft.

Note: Not to scale

α

β


06-100 ch 06 pp3 1/6/07 10:43 AM Page 684


6. Enter the equation you determined in Problem 5e into Y1 of your graphingcalculator. Choose a window that corresponds to the graph below. How doesyour graph compare to the graph below?

7. What is the largest value of the viewing angle? Justify your conclusion.

The largest viewing angle is approximately 3.7°. This is the output valueof the highest point on the graph.

8. How far should you stand from the mountain to obtain this maximum valueof the viewing angle? Explain.

I should stand approximately 470 feet from the mountain. This is theinput value for the largest viewing angle as seen on the graph.

1

2

3

4

5

6

250 500 750 1000 1250 1500x

f(x)

1750

Vie

win

g a

ng

le (

°)

Distance (ft.)


1. You are driving on a straight highway at sea level and begin to climb a hill witha 5% grade. (Recall that a grade is given in a percent but may be expressed as afraction. In that way you can view grade as a slope.)

a. What is the slope of the highway with a grade of 5%?

The slope is or or 0.05.

b. What is the angle of elevation, A? That is, what angle does the highwaymake with the horizontal? Explain.

The highway makes an angle of 2.86° with thehorizontal. This angle is called the angle of elevation.A � tan

�110.05 2 � 2.86°

120

5100

A

Note: Not to scale


06-100 ch 06 pp3 1/6/07 10:43 AM Page 685

c. If you walk along the highway for one mile, how many feet above sealevel are you? Explain.

1 mile is equivalent to 5280 ft. If x represents the number of feetabove sea level after walking 1 mile, then approximately 264 ft. I would be 264 ft. above sea level after 1 mile.

2. You are standing 92 meters from the base of the CN Tower in Toronto,Canada. You are able to measure the angle of elevation to the top of thetower as 80.6°.

a. Draw a diagram and indicate the angle of elevation.

b. What is the height of the tower? Round your answer to the nearest tenth.

556 meters because

3. You are in a spy satellite, equipped with a measuring device like a theodolite,orbiting five miles above Earth. Your mission is to discover the length of a se-cret airport runway. You measure the angles of depression to each end of therunway as 30° and 25°, respectively. What is the length of the runway?

Using the following diagram:

The two equations are

(a) and

(b) Solving

equation (b), miles.

Then equation (a) be-

comes

Solve this equationfor x.

miles The runway is approximately 2 miles long.x � 2.0

x �15 � 8.7 tan 25° 2

tan 25°

x tan 25° � 5 � 8.7 tan 25°

1x � 8.7 2 tan 25° � 5

tan 25° �5

x � 8.7.

y � 8.7

tan 30° �5y.

tan 25° �5

x � y

h � 92 tan 80.6° � 555.7 m

80.6°

h, height of tower

92 m

Top of tower

Me

x � 5280 # sin12.86° 2 �


BA

Spy satellite

5 miles

25° 30°

30°25°

x y miles

06-100 ch 06 pp3 1/6/07 10:43 AM Page 686


4. The Empire State Building rises 1414 feet above the ground, and you arestanding across 34th Street, approximately 80 feet from the base of thebuilding.

a. If you look up to the top of the building, what angle of elevation do youmake with the ground? Round your answer to the nearest degree.

The angle of elevation is or 87°.

b. How far from the building must you be to make your angle of elevationwith the ground 85°? Round your answer to the nearest foot.

Let d represent your distance from the Empire State Building.

ft.

I must stand 124 ft. from the base of the building for the angle ofelevation to be 85°.

d �1414

tan 85° � 124

d

85°

1414 ft

Note: Not to scale

u � tan�1 1141480 2 �

80 ft.

1414 ft.

Note: Not to scale

θ

06-100 ch 06 pp3 1/6/07 10:43 AM Page 687

5. Consider the function defined by defined for

a. Use your graphing calculator to sketch a graph of this function.

b. Over the given domain, what is the maximum value of the function, andwhere does it occur?

Over the given domain, the maximum value of the function isapproximately 19.5, and it occurs at

c. Over the given domain, what is the minimum value, and where does itoccur?

The minimum value is difficult to see on the graph itself. I used thetrace or the table feature on my graphing calculator to determinethat the minimum value in the given domain is 0.02865, and it occursat

6. You are interested in constructing a feeding trough for your cattle that canhold the largest amount of feed. You buy a 15 by 50-foot piece of aluminumto construct a 50-foot-long trapezoidal trough with a base of 5 feet. You bendup the two 5-foot sides through an angle of t° with the horizontal. Each crosssection is a trapezoid (see accompanying diagram). You need to determinethe angle, t, that produces the largest volume for the trough.

a. Recall the formula for the area of a trapezoid (see inside back cover of thetextbook, if necessary), and write the area of the trapezoidal cross sectionin terms of h and x (see accompanying diagram).

Let A represent the area of the trapezoidal cross section. The height ofthe cross section is h, and the two bases are 5 and respectively.

The area is then determined by the formula which,

after simplifying, is or A � 5h � hx.A � h15 � x 2

A �12 h110 � 2x 2,

5 � 2x,

xt

t

h

5

5

Note: Not to scale

t5 ft.

50 ft.

5 ft.5 ft.

Note: Not to scale

x � 0.01.

x � 14.

0.01 � x � 100.

f 1x 2 � arctan 20x � arctan

10x


06-100 ch 06 pp3 1/6/07 10:43 AM Page 688


b. Using right-triangle trigonometry, write h in terms of t. In a similar way,write x in terms of t. Using this information, write an equation for the areaof the trapezoid as a function of t.

or or

c. Use your graphing calculator to graph the area function you constructedin part b with Label your axes, and remember to indicateunits.

d. What angle produces the largest area for the trapezoid? Explain.

The graph in part c indicates that the area of the trapezoidal crosssection (output) is the greatest when the angle t is 60°.

e. What is that area? Explain.

The area is approximately 32.5 square feet as read from the graph inpart c.

f. Write the volume of the trough as a function of the angle t.

Let V represent the volume. Then,

where A is the cross-sectional area.

In terms of t, the volume is V � 1250 sin t11 � cos t 2

V � 50 # A,

5

10

15

20

25

30

35

10 20 30 40 50 60t°

Are

a (s

q. f

t.)

70 80 90

0° 6 t 6 90°.

1cos t 2sin tsin t � 25A � 25

A � 25 sin t11 � cos t 2A � 51sin t 2 15 � 5 cos t 2

x � 5 cos t

h � 5 sin t

06-100 ch 06 pp3 1/6/07 10:43 AM Page 689

g. Use your graphing calculator to graph the volume function you constructedin part f with Label axes and show your units.

h. What angle produces the largest volume, and what is that volume? Justifyyour conclusion.

The graph indicates the greatest value for the volume between 0°and 90° is approximately 1625 cubic feet when the angle t is 60°.

i. What do you conclude about the angle that produces the largest cross-sectional area and the angle that produces the largest volume for thetrough?

The angle is the same, namely 60° in this scenario.

250

500

750

1000

1250

1500

1750

10 20 30 40 50 60t°

Volu

me

(ft.

3 )

70 80 90

0° 6 t 6 90°.


06-100 ch 06 pp3 1/6/07 10:43 AM Page 690

1. Classical right-triangle trigonometry was developed by the ancient Greeksto solve problems in surveying, astronomy, and navigation. For purposes ofcomputation, the side opposite the angle , side O, is called the opposite, theside opposite the right angle, side H, is called the hypotenuse, and the thirdside, side A, is called the adjacent side to angle .

Define the three major trigonometric functions— , , and —interms of H, A, and O.

, ,

2. a. On any right length of the triangle, which trigonometric function wouldyou use to determine the length of the opposite side if you knew the anglemeasure and the length of the hypotenuse?

I would use the sine function.

b. Which trigonometric function would you use to determine the length ofthe adjacent side if you knew the angle measure and the length of the hy-potenuse?

I would use the cosine function.

c. Which trigonometric function would you use to determine the length ofthe adjacent side if you knew the angle measure and the length of the op-posite side?

I would use the tangent function.

d. Which trigonometric function would you use to determine the length ofthe opposite side if you knew the angle measure and the length of theadjacent side?

I would use the tangent function.

3. a. Suppose for a right triangle you know the length of the side opposite anangle and you know the length of the hypotenuse. How can you determinethe angle?

I would find the angle by using the arcsin function

b. There is another way to solve part a. Describe this alternative technique.

I could use the Pythagorean theorem to determine the length of the sideadjacent and then determine the angle by using the arccos function.

1sin�1 2.

tan u �OAcos u �

AHsin u �

OH

tan ucos usin u

θA

HO

u

u

CLUSTER 1 What Have I Learned?


06-100 ch 06 pp3 1/6/07 10:43 AM Page 691


4. If you know the lengths of two sides of a right triangle, how can you determineall the angles in the triangle? Make up an example, and determine all theangles. Remember that all the interior angles of a triangle add up to 180°.

Example:

Angle C is 90° by definition of a right triangle.

Then, and

Alternatively,


Calculate each of the following.

A

B

1 m

2 m

B � 90 � 36.9 � 53.1°.

B � tan�1 1432 � 53.1°.A � tan�1 1342 � 36.9°

A

B

C4

3

a. b. c.21

tan A 115

cos A 215

sin A

d. e. f.12

tan B 215

cos B 115

sin B

g. What trigonometric property is illustrated by parts a–f? Explain.

For complementary angles in a given right triangle, the cofunctionvalues of those angles are equal. Specifically, if A and B arecomplementary angles, then and

6. Consider the following two calculator screens.

sin B � cos A.sin A � cos B

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cluster 1 What Have I Learned? 693

a. The second screen indicates that 1.257 is not in the domain of the inversesine function. Do you agree? Explain.

Yes, I agree. Inputs are sine ratios, and the sine ratio is the length ofthe opposite side over the length of the hypotenuse. Because the op-posite is never longer than the hypotenuse, the ratio cannot exceed 1.

b. The following screen indicates that you do not have the same problem forthe inverse tangent function. Why not? Explain.

The ratios of can be much larger than 1,

because the length of the opposite can be larger than the length of

the adjacent; therefore, presents no problem.

7. Using diagrams, explain the difference between angle of depression andangle of elevation.

Angle of depression

Angle of elevation

y � 1.257

length of the opposite sidelength of the adjacent side

06-100 ch 06 pp3 1/17/07 3:48 PM Page 693


CLUSTER 1 How Can I Practice?

1. Triangle ABC is a right triangle.

Calculate each of the following. Write your answer as a ratio.

AC

B

15

178


a. b. c.1517

cos A158

tan B815

tan A

d. e. f.1517

sin B8

17

sin A817

cos B

2. Use your graphing calculator to determine the values of each of the following.Round your answers to the nearest thousandths.

a. 0.731 b. 0.574

c. 0.601 d. 5.671

3. Given determine and exactly.

and

4. Given determine and exactly.

and

5. Using your calculator, evaluate the following for to the nearest 0.1°, whereRound to the nearest tenth of a degree.0° � u � 90°.

u

cos B �4165sin B �

7165

cos Bsin Btan B �74,

tan A �5

12cos A �1213

tan Acos Asin A �5

13,

tan 80° �tan 31° �

cos 55° � sin 47° �

b.u � 23.5°

cos u � 0.9172 c.u � 74.1°

u � arctan 72

d.u � 16.6°

u � sin�1 27

a.u � 48.6°

sin u �34

e.u � 44.2°

u � tan�1 0.9714 f.u � 13.7°

u � arccos 0.9714

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cluster 1 How Can I Practice? 695

6. Solve the following right triangle. That is, determineall the missing sides and angles.

cm

cm

�C � 90°

�B � 37°

AC � 3.6

BC � 4.8

AC

B

53°

6.0 cm

7. You are building a new garage to be attached to your home and investigateseveral 30-foot-wide trusses to support the roof. You narrow down yourchoices to three: One has an angle of 45° with the horizontal, and the othershave angles of 35° and 25° with the horizontal. You determine that the wallsof the garage must be 10 feet high. To match the height of the rest of thehouse, the peak of the garage should be approximately 20 feet high. Whichtruss should you buy?

Therefore, I should buy the 35° trusses.

8. The side view of your swimming pool is shown here. The dimensions are infeet.

a. What is the angle of depression, ?

b. What is the length of the inclined side, D?

ft.D � 2162 � 62 � 17.1

u � tan�1 1 6162 � 20.6°

u

4

1010

10

36

θ

416

6D

A � arctan 110152 � 33.7°

15A

10

30

10

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9. a. As part of your summer vacation, you rent a cottage on a large lake. One

day, you decide to visit a small island that is 6 miles east and miles

north of your cottage. Draw a diagram for this situation. How far from

your cottage is the island?

The direct distance, d, from the cottage to the island is

mi.

b. At what angle with respect to due east should you direct your boat tomake the trip from your cottage to the island as short as possible?

I should direct my boat 22.6° north of east to get from the cottage tothe island in the shortest distance.

A � arctan 12.56 2 � 22.6°

d � 22.52 � 62 � 6.5

A

d, direct distance

6 mi.Cottage

2.5 mi.

Island

212


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Activity 6.6 Learn Trig or Crash! 697

ACTIVITY 6.6

Learn Trig or Crash!

OBJECTIVES

1. Determine the coordinatesof points on a unit circleusing sine and cosinefunctions.

2. Sketch a graph of y � sinx and y � cos x.

3. Identify the properties ofthe graphs of the sine andcosine functions.

CLUSTER 2 Why Are the Trigonometric Functions Called Circular Functions?

You are piloting a small plane and want to land at the local airport. Due to anemergency on the ground, the air traffic controller places you in a circular holdingpattern at a constant altitude with a radius of 1 mile. Because your fuel is low, youare concerned with the distance traveled in the holding pattern. Of course, youcommunicate with air traffic control about your coordinates so that you do notcollide with another airplane.

The given diagram shows your path in the air. The airport is located at the center10, 02 of the circle on the ground. The radius of the circle is 1 mile. A circlecentered at 10, 02 and having radius 1 is called a unit circle. You begin yourholding pattern at 11, 02.

Let’s examine the beginning of your first circular loop.

1. What distance (in miles) do you fly in one loop?

One loop is the circumference of the circle:mi.

2. Let P represent your position after flying only one-tenth of a loop. (Seepreceding diagram.)

a. Compute the distance traveled from 11, 02 to P.

mi.

b. Determine the number of degrees of the central angle, u, when you flyone-tenth of a loop (see preceding diagram). Recall that there are 360° ina circle.

The angle in Problem 2b is called a central angle.

110 360° � 36°

110 2p mi. �

p5 mi. � 0.628

C � 2p # 1 mi. � 2p mi. � 6.28

x

y

(1, 0)

P

θ1 mi.

DEFINITION

A central angle is an angle with its vertex at the center of a circle.

06-100 ch 06 pp3 1/6/07 10:43 AM Page 697


3. Now refer back to the right triangle in the preceding diagram. The measure ofthe central angle is 36°.

a. What is the length of the hypotenuse of the right triangle?

The hypotenuse is 1 mile long.

b. If 1x, y2 represents the coordinates of point P, then which coordinate, x ory, represents the length of the side opposite the 36° angle? Which letterrepresents the length of the side adjacent to the 36° angle?

y represents the length of the opposite side and x represents thelength of the adjacent side.

c. Use the appropriate trigonometric function to determine the value of x.

;

d. Use the appropriate trigonometric function to determine the value of y.

;

e. What are the coordinates of point P?

P 10.81, 0.592

Problem 3 demonstrates that if

a. an object is moving a distance d counterclockwise on the unit circle fromthe starting point 11, 02, and

b. P 1x, y2 represents the position of the object on the unit circle after it hasmoved distance d, and

c. represents the corresponding central angle,then the coordinates of P are given by

4. Repeat the procedure demonstrated in Problem 3 for the following fractionsof a loop, and record your results in the table.

(1, 0)(0, 0)

d

θ

P(x, y) = (cos , sin )θ θ

1cos u, sin u 2.u

y � 0.59sin 36° �y1

x � 0.81cos 36° �x1

72° mi. or 1.26 mi. 10.31, 0.952

45° mi. or 0.79 mi. 10.71, 0.712

18° mi. or 0.314 mi. 10.95, 0.3122p20

120

2p8

18

2p5

15

CENTRAL DISTANCE POSITION ON THELOOP ANGLE, uu TRAVELED CIRCLE 11cos uu, sin uu22

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Trigonometric Functions of Angles Greater than 90°5. Your instruments tell you that you have traveled 2 miles in the holding

pattern. What is the central angle, u? (Hint: What fraction of the loop haveyou traveled?)

2 miles means of a loop. The angle is.

For , the coordinates of P are now defined to be . Thisextends the idea of Problem 3 to larger angles, u. Depending on where you are onthe circle, these coordinates may be positive or negative.

6. a. From Problem 5, you know that the central angle is 114.6° (Seeaccompanying diagram). What is the measure of the central angle, u�,contained within the right triangle?

The measure of u� is

b. What is the length of the hypotenuse?

The length of the hypotenuse is 1 mile.

c. Using the sine and cosine functions, determine the lengths of the remainingtwo sides of the right triangle.

; the length of ; ; the length of

d. Using the results from part c, what are the coordinates of point P?Remember, the point is in quadrant II.

Since x is negative in quadrant II, the coordinates are 1�0.42, 0.912.

7. Your graphing calculator can be used to determine the coordinates of a point in adirect manner. It involves calculating the sine and cosine of the central angle u.

a. Determine the value of each of the following.

b. How do the results in part a compare to the coordinates of point P inProblem 6d?

The results are the same.

� 0.91sin 114.6°

� �0.42cos 114.6°

y � 0.91sin 65.4° �

y1x � 0.42cos 65.4° �

x1

180°�114.6° � 65.4°.

xx

y

y

(1, 0)= 114.6°θ

P(x, y) = (cos , sin )θ θ

1 mi.θ ′

1cos u, sin u 2u 7 90°

1p 360° � 0.321360° 2 � 114.6°

22p �

1p � 0.32

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8. Give an example of where on the unit circle (circle of radius 1) both coordinatesof P are negative. Place u and the coordinates of P on the following diagram.

(Answers will vary.)

9. a. By the time you have traveled 5 miles in the holding pattern, what is thecentral angle, u?

5 miles means of a loop, .

b. What are the coordinates of point P?

Graphs of Sine and Cosine Functions10. Complete the following table. Be sure your calculator is in degree mode.

Round the first column to the nearest hundredth and the third and fourthcolumns to the nearest thousandth.

P 1cos 286.5°, sin 286.5° 2 or approximately 10.28, �0.96 2

u �5

2p 360° � 286.5°52p

x

y

(1, 0)

(–0.95, –0.31) = 198°θ

(–0.71, –0.71) = 225°θ(–0.81, –0.59) = 216°θ

(–0.31, –0.95) = 252°θ

1 mi.

In general, the position P 1x, y2 of an object moving on a unit circle is given byand , where u is a central angle with initial side the positive

x-axis and terminal side OP, where O is the origin, the center of the circle.y � sin ux � cos u

DISTANCE TRAVELED CENTRAL ANGLE uu cos uu sin uu

0 0° 1 0

0.52 30° 0.866 0.5

0.79 45° 0.707 0.707

1.05 60° 0.5 0.866

1.57 90° 0 1

2.09 120° �0.5 0.866

3.14 180° �1 0

3.84 220° �0.766 �0.643

6 344° 0.961 �0.276

6.28 360° 1 0

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11. Plot the data pairs on the following grid. Draw a smooth curvethrough these points. Verify the results with your graphing calculator bysketching a graph of . Use the window Xmin � 0, Xmax � 360,Ymin � �1, and Ymax � 1 and degree mode.

12. Repeat Problem 11 for data pairs

13. You notice your instruments read 11.28 miles from the start of the holdingpattern. What is the central angle, u, and what are the coordinates of point P?

14. Suppose you are at some particular point in the holding pattern. How manymiles will you travel in the loop to return to the same coordinates?

or approximately 6.28 mi.

15. a. Complete the following table.

2p

P 1cos 646.3, sin 646.3 2 or approximately 10.28, �0.96 2

u �11.282p

# 360° � 646.3°

270° 360°

–1

90° 180°

1

θ

cosθ

1u, cos u 2.

–1

90° 180° 270° 360°

1

sin θ

θ

y � sin x

1u, sin u 2

uu (degrees) 0° 90° 180° 270° 360° 450° 540° 630° 720°

sin uu 0 1 0 �1 0 1 0 �1 0

cos uu 1 0 �1 0 1 0 �1 0 1

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b. Sketch a graph of for Verify using your graphingcalculator.

c. Sketch a graph of for

d. What pattern do you observe in each of the graphs?

(Answers will vary.) The patterns are wavelike and they repeat.

–1

θ90°

y

180° 270° 360° 450° 540° 630° 720°

1

0° � u � 720°.y � cos u

–1

θ90°

y

180° 270° 360° 450° 540° 630° 720°

1

0° � u � 720°.y � sin u

Note that these repeating graphs in Problem 15 show the periodic or cyclic behaviorof the trigonometric functions. Because many real-world phenomena involve thisrepeating behavior, the trigonometric functions are very useful in modeling thesephenomena.

DEFINITION

The shortest horizontal distance it takes for one cycle to be completed is calledthe period. The period is 360° for and .y � cos xy � sin x

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16. A negative angle is used to indicate that the object is moving along thecircumference of a unit circle in a clockwise direction.

a. Locate the point P that has a central angle of �30° on the followingdiagram.

b. Construct the appropriate right triangle. What is the measure of the centralangle within this right triangle? What is the length of the hypotenuse?

The measure of the central angle is 30°. The length of thehypotenuse is 1.

c. Use the sine and cosine functions to determine the lengths of the othertwo sides.

; the length of ; ; the length of

d. What are the coordinates of the point P?

Since P is located in quadrant IV, the coordinates of P are 10.87, �0.52.

e. Use your graphing calculator to determine and .How do these results compare to the coordinates of point P in part d?

The coordinates are the same.


sin 1�30°2cos 1�30°2

y � 0.5sin 30° �y1x � 0.87cos 30° �

x1

(1, 0)x

y

P1

30°

y

x

uu (degrees) cos uu sin uu

0° 1 0

�30° 0.87 �0.5

�90° 0 �1

�180° �1 0

�270° 0 1

�360° 1 0

06-100 ch 06 pp3 1/6/07 10:43 AM Page 703


b. Graph the points of the form using the values from the table inpart a.

c. Graph the points of the form using the values from the table inpart a.

d. Use your graphing calculator to graph and for

18. a. What is the domain of the sine and cosine functions?

The domain is all possible degree values.

b. What is the range of the sine and cosine functions?

The range is all real numbers between �1 and 1 including �1 and 1.

�720° � u � 720°.y � cos xy � sin x

–1

θ–90°

y

–180°–270°–360°

1

1u, cos u 2

–1

θ–90°

y

–180°–270°–360°

1

1u, sin u 2

06-100 ch 06 pp3 1/6/07 10:43 AM Page 704

1. A central angle is an angle whose vertex is the center of a circle.

2. The position P 1x, y2 of an object moving on the unit circle from the point 10, 12defines the sine and cosine functions by the rules , whereu is a central angle formed by the positive x-axis and the line segment OP.Because of this connection to the unit circle, the sine and cosine functions areoften called circular functions.

3. The domain of the sine and cosine functions is all angles, both positive andnegative.

4. The range of the sine and cosine functions is all values of N such that

5. The graphs (one cycle) of and look like the following.

sinx cosx

6. The period is 360° for and .y � cos xy � sin x

y � cos xy � sin x

�1 � N � 1.

y � sin ux � cos u,

SUMMARYACTIVITY 6.6



1. Determine the coordinates of the point on the unit circle corresponding to thefollowing central angles. Round to the nearest hundredth.

a.10.31, 0.952

b.10.64, �0.772

c.10, �12

270°310°72°

d.1�0.36, 0.932

e.1�0.85, �0.532

f.10.26, 0.972

435°212°111°

g.10.34, �0.942

2. For each of the points on the unit circle determined in Exercise 1, determine thedistance traveled from 11, 02 to the point along the circle. Round your answersto the nearest hundredth.

a. 72360

# 2p � 1.26

�70°

b. 310360

# 2p � 5.41

c. 270360

# 2p � 4.71

e. 212360

# 2p � 3.70

d. 111360

# 2p � 1.94

f. 435360

# 2p � 7.59


06-100 ch 06 pp3 1/6/07 10:43 AM Page 705


g.Therefore, the distance is 1.22.

3. Consider the graph of the following function.

a. Compare this graph to graphs studied in this activity.

The graph looks like the sine curve reflected in the t-axis.

b. What is the motion along the unit circle described by the graph?

The motion is clockwise rather than counterclockwise.

4. Consider the graph of the following function.

a. How does this graph compare to the graphs studied in this activity?

The graph looks like the cosine function reflected in the t-axis.

b. What is the motion around the unit circle described by the graph?

The motion indicates cosine values for point starting at 1�1, 02.P 1x,y 2

t

f (t)

1

1.5

0.5

–1

–1.5

–0.550 100 150 200 250 300 350

t

y

1.5

1

0.5

–1

–1.5

–0.550 100 150 200 250 300 350

�70360

# 2p � �1.22

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5. The following table represents the number of daylight hours for a certaincity in the Western Hemisphere on the dates indicated.

11.9 10.5 9.6 8.7 9.7 10.6 12.1 13.5 14.7 15.7 14.7 13.4 11.9

MAR. 21 APR. 21 MAY 21 JUNE 21 JULY 21 AUG. 21 SEPT. 21 OCT. 21 NOV. 21 DEC. 21 JAN. 21 FEB. 21 MAR. 21

a. Plot the data on the following grid.

b. Do the data indicate any similarities to a circular function (a functiondefined by points on the unit circle)? Explain.

Yes, the number of hours of daylight is cyclical. The graph looks like ashifted and stretched sine graph.

c. How does this function compare with the others in this activity?

The graph has the same wave-like shape.

d. Is the city in question north or south of the equator? Explain.

South; the number of hours of daylight is greater from October toFebruary, winter in the Northern Hemisphere.

////8

10

12

14

16

M A M J J AMonth

Ho

urs

of

Day

ligh

t

S O N D J F M

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Household electric current is called alternating current, or AC, because it changesmagnitude and direction with time. The household current through a 60-wattlightbulb is given by the equation

,

where A is the current in amperes and t is time in seconds.

Note that the input of the sine function in this activity is time measured by realnumbers (seconds) and not angles measured in degrees. In order for this to makesense, an alternate real-number method for measuring angles must be introduced.This method is called radian measure.

A � 2 sin 1120p t 2

ACTIVITY 6.7

It Won’t Hertz

OBJECTIVES

1. Convert between degreeand radian measure.

2. Identify the period andfrequency of a functiondefined by y � a sin 1bx2or y � a cos 1bx2 using thegraph.

Degree and radian measure correspond in the following way:

180° is the same as radians, or about 3.14 radians.

Therefore, dividing each side by 180° � radians by 180 gives you radi-

ans and dividing each side by p gives you 1 radian .�180°p

1° �p

1801p

1p

Whenever you see an angle measure without a degree symbol, assume that theangle is measured in radians.

Example 1 Dividing each side of the equality 180° � pp radians by 2 shows that

90 degrees is the same as radians. You can also convert 90 degrees

to radians by multiplying 1° by 90.

radians

Similarly, radians10° � 10

11pradians2

11802� 10 # p

180 radians �

p

18

�p

2 radians 90 # 1°.

90p180

radians�90° � 90 # 1° � 90 # p180

P

2

1. In the following table, convert degree measures to radian measure.

2. a. If you divide the equality 180° � p radians by p, you find that

1 radian . Then 2 radians would be , etc.

Generalize this to describe a procedure to convert radians to degrees.

Multiply the number of radians by .

b. How many degrees are there in 1.5p radians?

1.5p � � 270°1.5p # 180°p

180°p

2 180°p

�180°p

DEGREE MEASURE RADIAN MEASURE

10°

20°

30°

60°

120°

360° 2p � 6.283

2p3 � 2.094

p3 � 1.047

p6 � 0.524

p9 � 0.349

p18 � 0.175

06-100 ch 06 pp3 1/6/07 10:43 AM Page 708

Activity 6.7 It Won’t Hertz 709

c. How many degrees are there in 2p radians?

d. How many degrees are there in radians?

For more practice converting degree measure to radian measure and vice versa,see Appendix B.

Periodic Behavior of Graphs of Sine and Cosine FunctionsAs stated, the function defined by ,where A is the current inamperes and t is time in seconds, gives the household current through a 60-wattlightbulb.

3. Graph the equation using your calculator. Your calculatormust be in radian mode instead of degree mode. Using the indicated window,the graph should appear as follows:

Xmin � 0 Ymin � �3Xmax Ymin � 3

Xscl Yscl � 1�1

240

�1

20

A � 2 sin 1120p t 2

A � 2 sin 1120p t 2

p10

# 180°p � 18°

p10

2p # 180°p � 360°

BAppendix

4. Using the graph, determine the maximum current. Explain.

2 amps, the highest point represented on the graph; the lowest point,�2, represents 2 amperes in the opposite direction.

5. What do you think is happening to the current when the graph drops belowthe horizontal axis? (Reread the description of alternating current.)

It changes direction.

There is a pattern on the graph of starting at the 0 level, goingup to �2, down to �2, and returning to the 0 level. The pattern repeats. This patternis called a cycle.

Recall from Activity 6.6 that the shortest horizontal distance (on the input axis) ittakes for one cycle to be completed is called the period.

A � 2 sin 1120p t 2

Example 2 Determine the period of each of the following using its graph.

a. y � sin 12x 2 b. y � sin 112 x2

SOLUTION

a. b.

The screen in part a above indicates that the graph of y � sin 12x2 completes onecycle from 10, 02 to 1p, 02. Therefore, the period is �. The period in part b is 4�.

06-100 ch 06 pp3 1/6/07 10:43 AM Page 709

6. What is the period of the electric current function from Problem 3? (The tickmarks on the horizontal axis are at -second intervals.)

th sec.

7. Use the graph of each of the following functions to determine the period of thefollowing functions. Use the window Xmin � 0, Xmax � 6p, Xscl ,Ymin � �2, Ymax � 2, and Yscl � 1.

a.2�

y � sin x

�p4

160

1240

b.4�

y � sin 112 x2

c.2p3

y � cos 13x 2 d.3�

y � cos 123 x2

DEFINITION

The frequency of or is the number of cyclescompleted when the input, x, increases 2� units.

y � a cos 1bx 2y � a sin 1bx 2

Example 3 Determine the frequency of each of the following using its graph.

a. b.

SOLUTION

a.

The frequency is 2 because thegraph completes two completecycles in 2p units.

y � sin 112 x2y � sin 12x 2

b.

The frequency is 0.5 because thegraph completes half of onecycle in 2p units.

8. Determine the frequency, the number of cycles completed in 2� units, foreach of the following functions.

a.1

y � sin x b.12

y � cos 12 x c.

2

y � cos 2x


In general, the frequency of y � cos 1bx2 and y � sin 1bx2 is b.

9. For normal household current described by , how manycycles occur in one second?

The equation indicates that the current completes 120p cycles in 2p

seconds. So the current completes cycles in 1 second.120p2p

� 60

A � 2 sin 1120p t 2

06-100 ch 06 pp3 1/6/07 10:43 AM Page 710

1. Radian measure is used when the input of a repeating function is betterdefined by real numbers than angles measured in degrees.

2. Since 1° corresponds to radians, to convert degree measure to radian

measure, multiply the number of degrees by

3. To convert radian measure to degree measure, multiply the number of radians by

4. The pattern of a graph that covers all y-values once and is repeated is calleda cycle.

5. The smallest interval of input necessary for the graph of a function tocomplete one cycle is called the period.

Note: A formula for period will be developed in Activity 6.8

6. The frequency of a periodic or cyclic function is the number of cyclescompleted when the input is increased by 2p units.

180°p

.

pradians180°

.

p180

SUMMARYACTIVITY 6.7



1. Convert the following degree measures to radian measures.

a. 45°

radians45 # p180 �

p4 � 0.785

b. 140°


7p9 � 2.443

c. 330°


11p6 � 5.760

d. �36°

radians�36 # p180 �

�p5 � �0.628

2. Convert the following radian measures to degree measures.

a.3p4

# 180p � 135°

3p4 b. 2.5p

2.5p1

# 180p � 450°

c. 6p

6p # 180p � 1080°

d. 1.8p

1.8p # 180p � 324°

3. Complete the following table.

DEGREE MEASURE 0° 30° 45° 60° 90° 135° 180° 210° 270° 360°

RADIAN MEASURE 0 p 2p3p2

7p6

3p4

p

2p

3p

4P

6


For Exercises 4–9, be sure your calculator is in radian mode.

06-100 ch 06 pp3 1/6/07 10:43 AM Page 711



RADIANMEASURE, x 0 p 2p

sin x 0 0.707 1 0.707 0 �0.707 �1 �0.707 0

7p4

3p2

5p4

3p4

p

2p

4

b. Sketch a graph of using the table in part a.

c. Use your graphing calculator to sketch a graph of for.

5. How do the graphs of each pair of the following functions compare? Useyour graphing calculator.

a. ,

is stretched vertically by a factor of 2. is

compressed horizontally by a factor of 2.

b. ,

is stretched horizontally by a factor of 3.is

compressed horizontally by a factor of 3.y � cos x

y � cos 13x 2y � cos xy � cos 113 x2

y � cos 3xy � cos 13 x

y � cos x

y � cos 12x 2y � cos xy � 2 cos x

y � cos 2xy � 2 cos x

0 � x � 2pf 1x 2 � sin x

y = sin x

–1

p–4

p–2

p3p—4

3p—2

7p—4

2p5p—4

1

x

y

f 1x 2 � sin x

06-100 ch 06 pp3 1/6/07 10:43 AM Page 712


For Exercises 6–9, graph each function, and then determine the following foreach function:

a. The largest (maximum) value of the function

b. The smallest (minimum) value of the function

c. The period (the shortest interval for which the graph completes one cycle).

6.

max: 0.5

min:

period: p

�0.5

y = 0.5sin(2x)

–0.5

π–2

π–4

π3π—4

0.5

y � 0.5 sin 2x 7.

max: 3

min:

period: 2p3

�3

y = –3sin(3x)

–3

π–3

π–2

π–6

2π—3

3

y � �3 sin 3x

8.

max: 2.3

min:

period:

9. You are setting up a budget for the new year. Your utility bill for naturalgas and electric usage is a large part of your budget. To help determinethe amount you might need to spend on gas and electricity, you examineyour previous 3 years’ bills. Because you live in a rural area, you are billedevery 2 months instead of every month. The data you obtain appears in thefollowing graph.

4p

�2.3

f (x) = 2.3cos(0.5x)

2

1

–1

–2

–3–6 3 6

f 1x 2 � 2.3 cos 10.5x 2

06-100 ch 06 pp3 1/6/07 10:43 AM Page 713


You notice that a pattern develops, which is repeated. This function, thoughnot exactly periodic, models a periodic function for practical purposes. Labelthe horizontal axis with bimonthly periods, beginning with the June and Julybill from 3 years ago.

a. For what months of the year is the utility bill the highest? How much isthis bill?

The bill is highest for December and January. The amount of the bill isapproximately $600.

b. What months of the year is the utility bill the lowest? How much is thisbill?

The bill is lowest for June and July. The amount of the bill isapproximately $250.

c. What is the largest value for the function whose graph is given here?

The largest value is $650.

d. What is the period of the graph?

The period is 6 billing periods or 12 months.

e. Your power company announces that its rates will increase 5% beginningin April of the coming year. How will this change affect the graph of thisfunction? Will it affect the periodic nature of the function?

The graph will be stretched vertically by a factor of 1.05. This will notaffect the period of the function.

f. You heat your house and your water with natural gas and use electricityfor all other purposes. You do not currently have air-conditioning in yourhouse. If you were to install a central air-conditioning unit next summer,what changes do you think might occur in the shape of the graph?

The amount of the bills for the summer months would increase. Thegraph would flatten out as the monthly charges become more equal.

0M J S N J M M J S N J M M J S N J M M J S

100

200

Billing Period

Uti

lity

Bill

($)

300

400

500

600

700

Natural Gas and Electricity Usage

06-100 ch 06 pp3 1/6/07 10:43 AM Page 714

Activity 6.8 Get in Shape 715

You decide to try jogging to shape up. You are fortunate to have a large neighbor-hood park nearby that has a circular track with a radius of 100 meters.

1. If you run one lap around the track, how many meters have you traveled?Explain.

1 lap � the circumference of the circle

m

2. You start off averaging a relatively slow rate of approximately 100 meters perminute. How long does it take you to complete one lap?

min. More exactly, 2p minutes

You want to improve your speed. Gathering data describing your position on thetrack as a function of time may be useful. You sketch the track on a coordinatesystem with a center at the origin. Assume that the starting line has coordinates1100, 02 and that you run counterclockwise.

3. You begin to gather data about your position at various times. Using thepreceding diagram and the results from Problem 2, complete the followingtable to locate your coordinates at selected times along your path.

(100, 0)

y

x

t � 628 m100 m>min. � 6.28

C � 2pr � 2p 1100 m 2 � 628

ACTIVITY 6.8

Get in Shape

OBJECTIVE

1. Determine the amplitudeof the graph ofy � a sin 1bx2 or y � a cos 1bx2.

2. Determine the period ofthe graph of y � a sin 1bx2and y � a cos 1bx2 using aformula.

0 100 0

0 100

p �100 0

0 �100

2p 100 0

0 100

3p �100 0

5p2

3p2

p2

YOUR YOURt (min) x-COORDINATE y-COORDINATE

4. What patterns do you notice about the numerical data in Problem 3? Predictyour coordinates as your time increases.

The data is cyclical. For ; for t � 4p , 1x, y 2 � 1100, 0 21x, y 2 � 10, �100 2t �7p2 ,

06-100 ch 06 pp3 1/17/07 3:52 PM Page 715


5. To better analyze your position at times other than those listed in the previoustable, you decide to make some educated guesses. Let’s consider min-utes. Use the graph to approximate the coordinates of your position when

and label these coordinates on the graph.

6. To check your guess, you recall that right-triangle trigonometry gives someuseful information about special right triangles. (If you are not familiar withspecial right triangles, see Appendix B.) First, however, you need to computeu, given in the graph of Problem 5. Calculate u and explain how you arrive atyour answer.

There are two possible approaches: radians is equivalent to 45°, or arotation of units is 1/8 of a total rotation of 2p units; therefore, theangle is 1/8 of 360° or 45°.

7. On the following grid, use the data from Problems 3 and 6 to plot 1t, y2, youry-coordinate as a function of t. Connect your data pairs to make a smoothgraph, and predict what will happen to the graph for values of t before andafter the values of t in the tables.

100

–100

π–2

π–2

– π 3π—2

2π t

y

p4

p4

18 1360°2 � 45°

18 12p 2 �

p4

u �p4 � 45°

(100, 0)

(x, y) = (70.7, 70.7)p–4d = 100( ) = 25p

p–4

t =

25p——200p

1–8

= lap

100

y

xu

t �p4 ,

t �p4

BAppendix

06-100 ch 06 pp3 1/6/07 10:43 AM Page 716


8. Use your graphing calculator to plot . Make sure your graphingcalculator is in radian mode. Note that the name of the input t will need to bechanged to x to conform with the calculator.

9. Compare your answers to Problems 7 and 8.

The graphs are the same.

10. a. What is the maximum value of the sine function in Problem 8?

100 is the maximum value.

b. What is the minimum value of the sine function in Problem 8?

�100 is the minimum value.

Another important feature of the graphs of the sine and cosine functions is calledthe amplitude.

y � 100 sin 1t 2

DEFINITION

The amplitude of a periodic function equals

,

where M is the maximum output value of the function and m is the minimumoutput value of the function.

12 1M � m2

Example 1 The amplitude of the function defined by sin x is 1. Themaximum output value is 1. The minimum output value is � 1.The amplitude is 12 11 � 1�1 2 2 � 1

2 11 � 12 � 12 122 � 1.

y �

11. a. What is the amplitude of the sine function in the equation ?

100

b. Is there a relationship between the amplitude of and thecoefficient 100? Explain.

Yes; the coefficient 100 is the amplitude.

y � 100 sin x

y � 100 sin x

06-100 ch 06 pp3 1/6/07 10:43 AM Page 717


12. Determine by inspection the amplitude of the following functions. Thenverify your answers using your graphing calculator.

a.1.5

y � 1.5 sin x b.15

f 1x 2 � 15 sin 12x 2 c.3

y � 3 cos 113 x2

13. a. Is �2 the amplitude of ? Explain.

No. Amplitude is one-half the maximum minus the minimum, so itcan never be negative.

b. What is the amplitude of ?

2

c. Use your graphing calculator to sketch a graph of .

d. How does the graph of compare to the graph of ?

The graph of is the reflection of in the x-axis.

e. What is the general effect of the negative sign of the coefficient a in

The negative a will produce the graph of reflected in thex-axis.

y � a sin x

y � a sin x?

y � 2 sin xy � �2 sin x

y � 2 sin xy � �2 sin x

y � �2 sin x

y � �2 sin x

y � �2 sin x

In general, the amplitude of the functions defined by andin 0 a 0 .y � a cos1bx 2

y � a sin 1bx 2

Period of the Sine and Cosine FunctionsAfter a lot of practice, you begin to speed up on your circular track of radius100 meters.

You finally achieve your personal goal of 200 meters per minute.

14. a. If you run 200 meters per minute, how long does it take you to completeone lap? Note that this amount of time to complete one lap will be impor-tant in defining the key concept of period for the trigonometric functionsin the following problems.

one lap

min.t �200p m

200 m>min. � p min. � 3.14

� 2p 1100 m 2 � 200pm

06-100 ch 06 pp3 1/6/07 10:43 AM Page 718


b. Complete the following table to give your coordinates at selected special

points along your path. Since one lap takes p min, each of a loop

covers of the circular track.14

p4

YOUR YOURt (min) x-COORDINATE y-COORDINATE

0 100 0

0 100

�100 0

0 �100

p 100 0

0 100

0 100

0 100

2p 100 0

7p4

3p2

5p4

3p4

p2

p4

c. On the following grid, use the data from Problem 14b to plot 1t, y2, youry-coordinate as a function of t. Connect your data pairs to make a smoothgraph, as in the previous activity, and predict what will happen to thegraph for values of t before and after the values of t in the table.

The graph continues in the same pattern in both directions.

d. What is the amount of time it takes for the graph to complete one fullcycle (that is, for you to complete one full lap)?

� min.

e. What effect does doubling your speed have on the amount of time tocomplete one cycle? Explain.

Doubling the speed cuts the time in half.

–100

p–2

p–4

p 5p—4

7p—4

3p—4

3p—2

2p

100

t

y

06-100 ch 06 pp3 1/6/07 10:43 AM Page 719


15. a. Use your graphing calculator to plot . Note that the nameof the input t will need to be changed to x to conform with your calculator.

b. Compare the graph in part a to the graphics in Problem 14c.

The graphs are the same.

16. Recall that the period of a trigonometric function is the shortest interval ofinputs needed to complete one full cycle. Determine the period of each of thefollowing using the graph.

y � 100 sin12t 2

In general, if and , then the period is 2p >b.b 7 0y � a sin 1bx 2,

17. On the following grid, repeat Problem 14c to plot the data pairs 1t, x2.

18. a. What equation should you enter into your calculator to produce the graphfrom Problem 17?

y � 100 cos 12x 2

–100

p–2

p 3p—2

2p

100

t

x

a.period units

b.period units� p

y � 100 sin 12x 2

� 2p

y � 100 sin x

The period of the graph of a sine function can be determined directly from theequation that defines the function. For example, the period of is� units, which is half of the period of . It appears that thecoefficient of x in the function affects the period of the function.

y � 100 sin 1x 2y � 100 sin 12x 2

06-100 ch 06 pp3 1/6/07 10:43 AM Page 720


b. Enter the equation from part a into your calculator, and obtain a graph.

19. a. If you were to triple your speed on the track from the original 100 metersper minute, what effect would this have on the amount of time tocomplete one lap?

The time would be reduced by a factor of 3, from 2� min. (6.28 min.)to min. (2.09 min.).

b. What equations would describe the x- and y-coordinates of your position?

20. Determine the period of each of the following functions.

a.

b.

c.2p3

h1x 2 � 100 cos 13x 2

p

g1x 2 � 100 cos 12x 2

2p

y � 100 cos x

y � 100 sin 13t 2

x � 100 cos 13t 2

2p3

1. The amplitude of trigonometric functions defined by oris defined by

,

where M represents the maximum function value and m represents theminimum function value. The amplitude is equivalent to , the absolutevalue of the coefficient of the functions defined by and

. Therefore,

.

2. For the trigonometric functions defined by or ,

the period is , where

3. The frequency of trigonometric functions defined by oris the number of cycles completed by the graphs over

intervals of length . The frequency of trigonometric functions definedby or is b.y � a cos 1bx 2y � a sin 1bx 2

2py � a cos 1bx 2

y � a sin 1bx 2

b 7 0.2pb

y � a cos 1bx 2y � a sin1bx 2

0a 0 � 12 1M � m2


0a 0

12 1M � m2

y � a cos1bx 2y � a sin1bx 2SUMMARY

ACTIVITY 6.8

06-100 ch 06 pp3 1/6/07 10:43 AM Page 721


1. On the following grid, repeat Problem 16 to plot the data pairs 1t, x2.

2. Use your graphing calculator to plot . Make sure your graphingcalculator is in radian mode. Note that the names of both the input t and output(your x-coordinate) will need to be changed to x and y respectively conformwith your calculator.

3. a. What is the maximum function value of the cosine function in Problem 2?

The maximum value is 100.

b. What is the minimum function value of the cosine function in Problem 2?

The minimum value is �100.

4. What is the connection between the function values in Problem 3, the 100-meter radius of the circle, and the coefficient 100 of the function?

100 is the amplitude of the function.

5. If you are forced to run on a larger circular track of radius 150 meters, and youincrease your speed to 150 meters per minute, predict the equations of thefunctions describing the x- and y-coordinates of your position. Explain.

; y � 150 sin ux � 150 cos u

100

–100

p–2

p–2

– p 3p—2

2pt

y

x � 100 cos 1t 2

100

–100

p–2

p–2

– p 3p—2

2pt

x



06-100 ch 06 pp3 1/6/07 10:43 AM Page 722


6. Determine by inspection the amplitude of the following functions. Thenverify the results with your graphing calculator. (Remember: Amplitudecannot be negative.)

a.amp. � 3

b.amp. � 0.4

y � 0.4 cos xy � 3 sin x

c.amp. � 2

d.amp. � 2.3

g1x 2 � �2.3 sin xf 1x 2 � �2 cos x

e.amp. � 2

f.amp. � 4

h1x 2 � �4 cos 1x 2y � 2 sin 13x 2

7. Is there any relationship between the amplitude and the period of a sinefunction?No; they are independent.

8. For each of the following tables, identify a function of the form or that approximately satisfies the table.

a.

y � a cos bxy � a sin bx

x 0 0.7854 1.5708 2.3562 3.1416

y 0 �15 0 15 0

b.

y � �15 sin 12x 2

x 0 2.244 4.488 6.732 8.976

y 1.3 0 �1.3 0 1.3

9. For each of the following graphs, identify a function of the form y � a sin bxor y � a cos bx that the graph approximates.

a.

amp. 2; period or

b.

amp. 1.5; period , so that y � �1.5 cos 1px 2

b � p� 2 �2pb

x

y

2

–2

2–2 4 6

y � 2 sin 12x 2b � 2� p �

2pb

x

y

2

–2

2–2 4p 6

y � 1.3 cos 10.7x 2

06-100 ch 06 pp3 1/6/07 10:43 AM Page 723


10. Match the given equation to one of the graphs that follow. Assume thatXscl � 1 and Yscl � 1.

a.Graph is iii.

b.Graph is iv.

y � �0.5 sin 2xy � 2 cos 0.5x

c.Graph is i.

d.Graph is ii.

y � 2 sin 0.5xy � 0.5 cos 2x

i. ii.

iii. iv.

06-100 ch 06 pp3 1/6/07 10:43 AM Page 724

Activity 6.9 The Carousel 725

You have developed a keen interest in carousels. You developed enough of aninterest to visit six carousel parks on your vacation.

At one of the parks you are watching a carousel and notice a young girl andher brother as they rush to pick out their special horses for their rides. Thegirl chooses one right in front of you. Her brother chooses one in the same rowonly of the way around the carousel, and of course ahead of his sister. You de-cide to investigate their relative positions as they enjoy their rides.

Each child is 20 feet from the center of the carousel. Assume the girl starts at thepoint 120, 02 and her brother starts at 10, 202.

1. Complete the following table to give coordinates for both the girl and herbrother at selected special points on their ride.

14

ACTIVITY 6.9

The Carousel

OBJECTIVE

1. Determine thedisplacement ofy � a sin 1bx � c2 andy � a cos 1bx � c2using a formula.

t (AMOUNT THE GIRL’S THE GIRL’S HER BROTHER’S HER BROTHER’SOF ROTATION x-COORDINATE y-COORDINATE x-COORDINATE y-COORDINATEIN RADIANS)

0 20 0 0 20

0 20 �20 0

� �20 0 0 �20

0 �20 20 0

2� 20 0 0 20

0 20 �20 05p2

3p2

p2

2. On the following grid, use the data from Problem 1 to plot 1t, y2 for both thegirl and her brother. Connect the points to smooth out your graphs.

3. What is the relationship between the two graphs?

The graph for the boy is shifted units to the left of the graph for thegirl.

p2

Her brother

–20

π–2

π 3π—2

2π 5π—2

20

Thegirl

t

06-100 ch 06 pp3 1/6/07 10:43 AM Page 725


DEFINITION

The displacement, or phase shift, of the graph of is thesmallest movement (left or right) necessary for the graph of tomatch the graph of exactly.y � a sin 1bx � c 2

y � a sin 1bx 2y � a sin 1bx � c 2

Example 1 Consider the graphs of y � sin x and y � sin 11x � 122 .

The graph of must be moved 1 unit to the left to match the graph ofexactly, so the displacement, or phase shift, is �1.

4. a. Which graph is displaced in Problem 2?

The graph for the boy.

b. What is the displacement?

units to the left.

5. Would you expect the same type of relationship between the two graphsrepresenting the x-coordinates? Explain.

Yes, you can see from the table that the x-coordinates are the same forvalues of t that differ by units.

Displacement, or phase shift, is defined for the cosine function in the samemanner it is defined for sine.

Example 2 Consider the graphs of y � 3 cos 2x and y � 3 cos 112x � 122 .

The graph of appears to be about unit to the right of the

graph of , so the displacement is approximately .12y � 3 cos 2x

12y � 3 cos 12x � 1 2

y = 3 cos2x y = 3 cos (2x – 1)2

–2

0.5–0.5–1–1.5 1 1.5

p2

p2

y � sin 1x � 1 2y � sin x

y = sin xy = sin(x + 1)

1

–1

1–1–2–3 2 3

06-100 ch 06 pp3 1/6/07 10:43 AM Page 726


6. a. If the girl’s x-coordinate is given by 20 cos t, predict the defining equationof the boy’s x-coordinate.

b. Use your graphing calculator to test your prediction.

x � 20 cos 1t �p2 2

In general, in the functions defined by and the values for b and c affect the displacement, or phase shift, of the function. Thephase shift is given by . Note that if is negative, the shift is to the left. If is positive, the shift is to the right.

�cb�

cb�

cb

cos 1bx � c 2,y � ay � a sin 1bx � c 2

7. a. Using the expression , what is the displacement of ?

b. Is your result consistent with the graph in Example 2?

Yes.

8. In this activity, the boy’s y-coordinate is given by , whereand . Calculate c.

c � �p2

�c1 �

�p2

�cb �

p�2

b � 1a � 20y � a sin 1bx � c 2

�cb � �

1�1 22 �

12

y � 3 cos 12x � 1 2�cb

1. The displacement, or phase shift, of the graph of ,, is the smallest movement (left or right) necessary for the graph of

to match the graph of exactly.

2. The displacement, or phase shift, of the graph of ,, is the smallest movement (left or right) necessary for the graph of

to match the graph of exactly.

3. For the functions and , , thephase shift is given by .

4. If is negative, the shift is to the left. If is positive, the shift is to theright.

�cb�

cb

�cb

b 7 0y � a cos 1bx � c 2y � a sin 1bx � c 2

y � cos 1bx � c 2y � a cos 1bx 2b 7 0

y � a cos 1bx � c 2

y � a sin 1bx � c 2y � a sin bxb 7 0

y � a sin 1bx � c 2SUMMARYACTIVITY 6.9

06-100 ch 06 pp3 1/6/07 10:43 AM Page 727


c.amplitude: 2.5

period: 5�

displacement: �p3

0.4� �

5p6

f 1x 2 � �2.5 sin 10.4x �p3 2 d.

amplitude: 15

period: 1

displacement: �1�0.3 2

2p �3

20p � 0.0477

g1x 2 � 15 sin 12px � 0.3 2

2. For each of the following tables, identify a function of the formor that approximately satisfies the

table.

a.

y � a cos 1bx � c 2y � a sin 1bx � c 2

x �0.7854 0.7854 2.3562 3.927 5.4978

y 0 3 0 �3 0

b.

y � 3 sin 1x �p4 2

x 1 2.5708 4.1416 5.7124 7.2832

y �0.5 0 0.5 0 �0.5

3. Sketch one cycle of the graph of the function defined by

f(x) = 2sin(x + )2

–2

p–2

p–2

– p 3p—2

p–2

x

f(x)

f 1x 2 � 2 sin 1x �p2 2 .

y � �0.5 cos 1x � 1 2


1. For the following functions, identify the amplitude, period, and displacement.

a.amplitude: 0.7

period: �

displacement: �p2

2� �

p4

y � 0.7 cos 12x �p2 2 b.

amplitude: 3

period: 2�

displacement: �1�1 2

1 � 1

y � 3 sin 1x � 1 2


06-100 ch 06 pp3 1/6/07 10:43 AM Page 728


5. Match the given equation to one of the graphs that follow. (Assume thatXscl � 1 and Yscl � 1.)

a.Graph is iii.

b.Graph is iv.

y � 2 sin 1x � 1 2y � 2 cos 1x � 1 2

c.Graph is i.

d.Graph is ii.

y � 2 sin 1x � 2 2y � 2 cos 1x � 2 2

i. ii.

iii. iv.

4. Determine an equation for the function defined by the following graph.

y � 2 sin 1x � 1 2

x

y

2

1

–2

–121 3 5–1 4 6

06-100 ch 06 pp3 1/6/07 10:43 AM Page 729


According to the National Weather Service, the average monthly high temperaturein the Midland-Odessa, Texas, area between 1971 and 2000 is given by thefollowing table.

ACTIVITY 6.10

Texas Temperatures

OBJECTIVES

1. Determine the equationof a sine function thatbest fits the given data.

2. Make predictions using asine regression equation.

MONTH Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

AVER.MAX. 56.8 63.0 70.9 78.8 86.8 92.7 94.3 92.8 86.1 77.4 65.9 58.4

1. To get a feel for the relationship between the month and the averagehigh temperatures, let x represent the month, where 1 � Jan of the first year,12 � Dec of the first year, 13 � Jan of the second year, and 24 � Dec of thesecond year. Plot the average maximum monthly temperatures for Midland-Odessa over a 2-year period on the grid below.

2. Enter the data plotted in Problem 1 into your calculator, and verify your scatter-plot by creating a stat plot. Your plot should resemble the screen below.

Notice the wavelike and repetitive nature of the data. This is typical of manynatural periodic phenomena.

50

60

70

80

90

100

y

x3 6 9 12 15 18 21 24

Month

Ave

rage

Max

. Tem

pera

ture

06-100 ch 06 pp3 1/6/07 10:43 AM Page 730

Activity 6.10 Texas Temperatures 731

To produce a sine function that models the data, set your calculator to radian modeand use the Stat Calc menu and Option C: SinReg.

Note that the sine regression model is of the form

y � a sin 1bx � c2 � d

3. Enter your regression model here, rounding the values of a, b, c, and d to thenearest 0.001

y � 18.614 sin 10.521x � 1.9842 � 76.900

4. Add the graph of this function to your stat plot to verify that it is a goodmodel. Your screen should appear as follows.

1. The sine function can be used to model periodic wavelike behavior.

2. To determine the sine regression using the TI-83/TI-84 Plus calculator,

a. set your MODE to radian, and

b. use the SinReg option of the Stat Calc menu.

SUMMARYACTIVITY 6.10


The phases of the Moon are periodic and repetitive even though the names of thephases can be confusing.

i. A new Moon occurs when no moon is visible to an observer on Earth.

ii. During the first quarter, half of the side of the Moon facing Earth is visible.

iii. During a full Moon, the entire side of the moon is visible.

iv. During the last quarter, the other half of the side of the Moon facing Earth isvisible.


DATE Jan 29 Feb 5 Feb 13 Feb 21 Feb 28 Mar 6 Mar 14 Mar 22 Mar 29

MOON 1st Last 1st LastPHASE

New quarter Full quarter New quarter Full quarter New

The U.S. Naval Observatory lists the following dates for phases of the Moon inthe year 2006.

06-100 ch 06 pp3 1/6/07 10:43 AM Page 731


1. a. Let x represent the number of days since Jan 29 (be careful here) and let yrepresent the amount of moon visible, where New � 0, the 1st quarter �0.5, Full � 1, and the Last quarter � 0.5. Complete the following table.

DATE Jan 29 Feb 5 Feb 13 Feb 21 Feb 28 Mar 6 Mar 14 Mar 22 Mar 29

x; DAYS SINCEJAN 29 0 7 13 21 28 34 42 50 57

y; THE AMOUNTOF MOON VISIBLE 0 0.5 1 0.5 0 0.5 1 0.5 0

b. Plot the data points generated in the table in part a.

c. Does this data indicate that a sine regression may be appropriate in thiscase? Explain.

Yes. It does show repeated and periodic behavior.

d. Use your TI-83/TI-84 Plus calculator to produce a sine regression modelfor the data in part a. Round a, b, c, and d to the nearest 0.001 and recordthe model below.

y � 0.500 sin 10.222x � 1.5092 � 0.521

e. Use your model to predict approximately how much of the Moon will bevisible 95 days after Jan 29th, provided the sky is clear.

y1952 � 0.85, or about 85% of the side of the Moon facing the earthwill be visible 95 days after Jan 29th.

0.5

1

1.5

2.5

2

0 9 18 27 36 45 54 63 72

Days Since Jan. 29

Am

ount

of

Moo

n V

isib

le

x

y

06-100 ch 06 pp3 1/6/07 10:43 AM Page 732

Activity 6.10 Texas Temperatures 733

2. Tides are the regular rising or falling of the ocean’s surface. This is due inlarge part to the gravitational forces of the Moon. The following tablerepresents water level of the tide off the coast of Pensacola, Florida, for a24-hour period, January 2 and 3 of 2006.

HOUR # 0 2 4 6 8 10 12 14 16 18 20 22 24

Measurement in feetabove average low tide �.24 �.07 .29 .66 1.02 1.27 1.32 1.07 .66 .2 �.16 �.47 �.53

Data source: NOAA tidesonline.nos.noaa.gov/

a. Plot the data points given in the table above using hour as the independentvariable.

b. Does this data indicate that a sine regression may be appropriate in thiscase? Explain.

Yes, the wavelike behavior is apparent in the scatterplot.

c. Use your TI-83/TI-84 Plus calculator to produce a sine regression modelfor the data in the table. Round a, b, c, and d to the nearest 0.001 andrecord the model below.

y � 0.884 sin 10.245x � 1.0932 � 0.400

d. Use your model to predict the height of the tide 30 hours after the originalobservation.

y 1302 � 0.380 ft. above the average low tide

e. Do you expect your model from part c to be a good predictor of tideheight on a year-round basis?

No, the height of the tide is affected by weather, so the wavelikebehavior should be there but the heights will vary. (Students mayalso know that the tide is affected by the Moon and its rotationperiod is about 29.5 days.)

20.60

1.00

2.00

0 2 4 6 8 10 12 14 16 18 20 22 24

Hour Number

Wat

er H

eigh

t Abo

ve A

vera

ge L

ow T

ide

x

y

06-100 ch 06 pp3 1/6/07 10:43 AM Page 733


Sounds are all around us. Vibrations create waves moving through air, whichwhen received by our eardrum result in our hearing a sound. Music is composedof tones that are made of evenly spaced waves that move through the air at thespeed of sound. Higher and lower tones (the pitch of a note) are caused bydifferent spacing in the waves. The closer together the waves are, the higher thetone sounds. This spacing of the waves, the distance from one high point of awave to the next wave’s high point, is called the wavelength. It is also the periodof the underlying sine curve, the shape of the wave.

Since all sound waves are traveling at about the same speed, the speed of sound,waves with a longer wavelength don’t vibrate your ear as often (frequently) as theshorter waves. This aspect of a sound, how often a wave goes by, is calledfrequency by scientists and engineers. They measure it in hertz (Hz), which is thenumber of waves that go by per second. People can hear sounds that range fromabout 20 to about 17,000 Hz.

1. Sound travels at about 344 meters per second in air (at 20°C). What is thelongest wavelength that people can normally hear?

Hint:

17.20 m

2. What is the shortest wavelength that people can normally hear?

0.020 m

In music, frequency is called the pitch of a sound. The shorter the wavelength, thehigher the frequency, and hence the higher the pitch of the sound. Instead ofmeasuring frequencies, musicians name the pitches by referring to notes on thetreble or bass clef scales. For example, the frequency of the note called middle Cis about 262 Hz.

3. What is the wavelength of middle C?

1.313 m

Some note combinations sound pleasing to our ears. To understand why somenote combinations sound better than others, let’s first look at the wave patterns oftwo notes that are one octave apart.

bmeterssecond

# secondwave

�meterswave

a

Higher tone

one wavelength

Lower tone

PROJECTACTIVITY 6.11

Music and Harmony

OBJECTIVES

1. Know the relationshipbetween wavelength andfrequency.

2. Determine the sine modelfor a given frequency.

06-100 ch 06 pp3 1/6/07 10:43 AM Page 734

Activity 6.11 Music and Harmony 735

4. What is the ratio of the wavelengths of these two notes?

2 to 1

The following wave pattern represents two notes that would sound terrible together.

Note that the two waves never seem to intersect on the midline except at the verybeginning of both waves. They are truly “out-of-sync,” or dissonant in musicalterms. Whenever two waves come together after several wavelengths to form arepeating pattern, a harmonious sound pleasing to the ear is heard. Musicians callthis phenomenon consonance.

5. What is the ratio of wavelengths in the wave pattern below?

4 to 3 (or 3 to 4)

6. What is the ratio of the frequencies of these two waves?

3 to 4 (or 4 to 3)

A chord consists of notes that sound good together. The C-major chord, starting atmiddle C, has the following frequencies:

C—262 Hz

E—330 Hz

G—392 Hz

7. Determine the ratios of the frequencies of E to C, G to C, and G to E.

E to C : 1.260; G to C : 1.496; G to E : 1.188

06-100 ch 06 pp3 1/6/07 10:43 AM Page 735


8. Due to rounding, the ratios may not be exactly simple integer ratios (like 3to 2), but they are very close. Express your ratios in Problem 7 as simple inte-ger ratios.

E to C : 1.260 � 1.25 � 5 to 4

G to C : 1.496 � 1.5 � 3 to 2

G to E : 1.188 � 1.2 � 6 to 5

9. Remembering that the ratios in Problem 8 are ratios of the frequencies,

a. how many E waves will fit in the length of four C waves?

5

b. how many G waves will fit in the length of four C waves?

6

To verify your answers to Problem 9, you can graph a sine curve model for each

of the waves. Recall that the waves will have a period � , so .

10. You determined the wavelength (period) of middle C in Problem 3. Graph themiddle-C wave over four periods (wavelengths). Use the window

.

Record the equation for your model.

, so y � sin 14.785x 2b �2p

1.313� 4.785

c0 � x � 4 # period�2 � y � 2

d

b �2p

period2pb

11. Note that the amplitude for your model is 1. What do you think a change inamplitude would represent for this note, or any other?

The greater the amplitude, the louder the note.

12. Now determine the equation for the sine curve that models the E wave. Graphthe E wave with the middle-C wave and verify your answer to Problem 9.

wavelength of m

, so y � sin 16.03x 2b �2p

1.042� 6.03

E �344330

� 1.042

06-100 ch 06 pp3 1/6/07 10:43 AM Page 736


1. The C-major key, starting with middle C, consists of seven notes (white keyson the piano) with the following frequencies.


DATE C D E F G A B

FREQUENCY (Hz) 261.6 293.7 329.6 349.2 392.0 440.0 493.9

a. Determine the ratio of each note’s frequency to middle C.

D to C � 1.1224 G to C � 1.4983

E to C � 1.2599 A to C � 1.6818

F to C � 1.3348 B to C � 1.8877

b. None of the ratios is a perfect simple ratio (ratio of small counting numbers),but most are close. Determine a simple ratio that is close to the ratios you foundin part a. (You already identified two of them in Problem 8 of the activity.)

D to C G to C

E to C A to C

F to C B to C

c. The smaller the numbers in the ratio, the greater the consonance. Which notewhen played with middle C will have the greatest consonance? Which notewhen played with middle C will have the greatest dissonance? (If you haveaccess to a keyboard, try playing each of the pairs simultaneously, to see ifyour ear agrees with your answer.)

greatest consonance: G & C

greatest dissonance: B & C

2. When a string on a guitar or violin is plucked, the vibration creates a wave thatproduces a sound. The length of the string (along with its thickness, tension, anddensity) will produce a particular note. The study of vibrating strings led to muchof the current Western music tradition. The Greek mathematician Pythagoras wasintrigued with the notes created by a vibrating string. He noticed that the toneproduced by halving the length of a string will be the same tone (as harmoniousas possible) but one octave higher. The wavelength is exactly half the original.

a. The frequency of middle C is 262 Hz. What will be the frequency of C oneoctave above and one octave below middle C?

one octave above middle C � 524 Hz

one octave below middle C � 131 Hz

�1.8877 � 1.888 p � 17>9�1.3348 � 1.333 p � 4>3

�1.6818 � 1.666 p � 5>3�1.2599 � 1.25 � 5>4

�1.4983 � 1.5 � 3>2�1.1224 � 1.125 � 9>8

Activity 6.11 Music and Harmony 737

06-100 ch 06 pp3 1/6/07 10:43 AM Page 737


b. If the string is depressed one-third from the end, the resulting notes are inthe ratio 3 to 2. This is called a perfect fifth. In the C-major key, whichtwo notes form a perfect fifth?

G and C

3. When an octave is divided into 12 equal steps, a chromatic scale results. Butthe steps are in terms of ratios of the frequencies. In other words, the ratiosbetween successive notes are constant. Recall middle C has a frequency of261.6 Hz. The C note one octave higher has twice the frequency, 523.2 Hz.Each of the 11 notes in between can be determined by multiplying by thecommon ratio, r, starting with middle C. So the next higher C will have afrequency equal to r121261.62 � 523.2.

a. Solve this equation for r, and approximate to four decimal places.

b. The 12 notes include all the so-called sharp (or flat) notes, the black keyson the piano keyboard. C# is the symbol for C sharp, one step higher thanC. Determine the frequencies for the chromatic scale.

r � 211�122 � 1.059463

261.6 277.2 293.6 311.1 329.6 349.2 370.0 392.0 415.3 440.0 466.1 493.8 523.2

C C# D D# E F F# G G# A A# B C

c. What is the ratio of frequencies between G# and D#? Is it close to a ratioof simple and low counting numbers?

, so the notes should be consonant.

d. Determine the wavelengths for D# and G#, and a sine model for each.Graph over four periods to see if the notes appear to be dissonant orconsonant.

D# : ; so

E# : ; so

Appears consonant:

y � sin 17.59x 2b �2p

0.828� 7.59

344415.3

� 0.828 mi;

y � sin 15.68x 2b �2p

1.106� 5.68

344311.1

� 1.106 mi;

415.3311.1

� 1.33494 � 1.33 p �43

06-100 ch 06 pp3 1/6/07 10:43 AM Page 738

Cluster 2 What Have I Learned? 739

1. Explain why the trigonometric functions could be called circular functions.

Their output values are coordinates of points on a circle.

2. Sometimes the difference between the trigonometric functions and thecircular functions is explained by the difference in inputs. The input valuesfor the trigonometric functions are angle measurements, and the input valuesfor the circular functions are real numbers. How does your knowledge ofradian measure relate to this?

An angle measured in radians is equivalent to the distance traveledalong an arc of a circle of radius 1 intercepted by the angle. Thisdistance is a real number.

3. a. Estimate the amplitude of the function defined by the following graph.

The amplitude is about 6.

b. Use the definition of amplitude to answer part a.

4. The period of and is 2p. Explain why this makes sensewhen sine and cosine are viewed as circular functions.

The circumference of a circle is 2�r. If the radius is 1, then the length ofthe intercepted arc is 2� units.

5. Given a function defined by and , and given thefact that b and c possess opposite signs, determine whether the graph of thefunction is displaced to the right or to the left. Explain.

If b and c have opposite signs, then is negative and is positive.Therefore, the displacement is to the right.

�cb

cb

b 7 0y � a sin 1bx � c 2

y � cos xy � sin x

amp. �12 1M � m 2 �

12 149 � 37 2 �

12 112 2 � 6

44

36

46

48

50

y

38

40

42

CLUSTER 2 What Have I Learned?

06-100 ch 06 pp3 1/6/07 10:43 AM Page 739


1. Determine the coordinates of the point on the unit circle corresponding tothe following central angles. If necessary, round your results to the nearesthundredth.

a. 36°

10.81, 0.592

CLUSTER 2 How Can I Practice?

Answers to all How Can I Practice exercises are included in the Selected Answers appendix.

b. 210°

1�0.87, �0.52

c.10, �12

�90° d.10.73, �0.682

317°

e.1�0.81, �0.592

�144° f.10, 12

450°

2. For each of the points on the unit circle determined in Exercise 1, determinethe distance traveled along the circle to the point from 11, 02.

a. 0.63 units b. 3.67 units

c. 1.57 units clockwise d. 5.53 units

e. 2.51 units clockwise f. 7.85 units

3. Convert the following degree measures to radian measures in terms of p.

a. 18°18 # p

180 �p10

b. 150°

150 # p180 �

5p6

c. 390°

390 # p180 �

13p6

d. �72°

�72 # p180 �

�2p5

4. Convert the following radian measures to degree measures.

a.5p6

# 180p � 150°

5p6 b. 1.7p

1.7p # 180p � 306°

c. �3p

�3p # 180p � �540°

d. 0.9p

0.9p # 180p � 162°

06-100 ch 06 pp3 1/6/07 10:43 AM Page 740

Cluster 2 How Can I Practice? 741

Obtain the following information about each of the functions defined by theequations in Exercises 5–9.

a. Use your graphing calculator to sketch a graph.

b. From the defining equation, determine the amplitude. Then use yourgraph to verify that your amplitude is correct.

c. From the defining equation, determine the period. Then use your graph toverify that your period is correct.

d. From the defining equation, determine the displacement. Then use yourgraph to verify that your displacement is correct.

5.amplitude: 4

period:

displacement: 0

2p3

y � 4 cos 3x 6.amplitude: 2

period:

displacement: 1

2p1 � 2p

y � �2 sin 1x � 1 2

7.amplitude: 3.2

period:

displacement: 0

2p2 � p

s � 3.2 sin 1�2x 2 8.amplitude: 1

period:

displacement: �2

2p12

� 4p

f 1x 2 � �cos 1x2 � 12

9.amplitude: 3

period:

displacement: �1�1 2

4 �14

2p4 �

p2

g1x 2 � 4 � 3 cos 14x � 1 2

06-100 ch 06 pp3 1/6/07 10:43 AM Page 741


10. You rent a cottage on the ocean for a week one summer and notice that thetide comes in twice a day with approximate regularity. Remembering thatthe trigonometric functions model repetitive behavior, you place a meter-stick in the water to measure water height every hour from 6:00 A.M. to mid-night. At low tide the height of the water is 0 centimeters, and at high tidethe height is 80 centimeters.

a. Explain why a sine or a cosine function models this relationship betweenheight of water in centimeters and time in hours.

A sine or cosine function models this relationship because of therepetitive nature of the height of the water as a function of time.

b. What is the amplitude of this function?

amplitude

c. Approximate the period of this function. Explain.

The period is approximately 12 hours, because high tide occurs twicea day.

d. Determine a reasonable defining equation for this function. Explain.

Let x represent the number of hours since midnight.

, the amplitude

period:

displacement:

vertical shift:

Other equations are possible; depends on the choice of thedisplacement.

y � 40 sin 1 p6 x �p2 2 � 40

d � 40

c ��p

2

�c � 3 # 1 p6 23 �

�cb �

�cp6

b �p6

2pb � 12

a � 40

y � a sin 1bx � c 2 � d

�80 � 0

2� 40

06-100 ch 06 pp3 1/6/07 10:43 AM Page 742

743

SummaryC h a p t e r 6

The bracketed numbers following each concept indicate the activity in which the concept is discussed.

CONCEPT/SKILL DESCRIPTION EXAMPLE

The sine function of the acute angleA of a right triangle [6.1] sin A �

length of the side opposite Alength of the hypotenuse

Examples 2 and 3, Activity 6.1,pages 659–660

Solving right triangles [6.4] When trigonometric problems involvesolving right triangles, employ thefollowing trigonometric problem-solvingstrategy.

1. Draw a diagram of the situation usingright triangles.

2. Identify all known sides and angles.3. Identify sides and/or angles you want

to know.4. Identify functions that relate the

known and unknown.5. Write and solve the appropriate

trigonometric equation(s).

Example 1, Activity 6.4,page 678

The tangent function of the acuteangle A of a right triangle [6.1] tan A �

length of the side opposite A

length of the side adjacent to A


Cofunctions related tocomplementary angles [6.2]

Cofunctions of complementary anglesare equal.

sin 35° � cos 55°

The domain of the inversetrigonometric functions [6.3]

The domain (set of inputs) of the inversetrigonometric functions is the set ofratios of the lengths of the sides of aright triangle.

The domain of the inverse sinefunction is all real numbers from�1 to 1, including both �1 and 1.

The cosine function of the acuteangle A of a right triangle [6.1] cos A �

length of the side adjacent to Alength of the hypotenuse


Complementary angles [6.2] Complementary angles are two acuteangles, the sum of whose measures is 90°.

Angles of 30° and 60° arecomplementary angles.

Inverse trigonometricfunctions [6.3]

The inverse trigonometric functions aredefined by

is equivalent to

is equivalent to

is equivalent to tan u � xtan �1x � u

cos u � xcos �1x � u

sin u � xsin �1x � u


The range of the inversetrigonometric functions [6.3]

The range (set of outputs) of the inversetrigonometric functions is the set ofangles.

The range of the inverse tangentfunction is all angles from to 90°.

�90°

06-100 ch 06 pp3 1/6/07 10:43 AM Page 743

744


The amplitude of trigonometricfunctions [6.8]

The amplitude of trigonometricfunctions defined by or

is defined by

,

where M represents the maximumfunction value and m represents theminimum function value.

12

1M � m2


Example 1, Activity 6.8, page 717

Converting from radian measure todegree measure [6.7]

To convert degree measure to radianmeasure, multiply the degree measure by

.p radians

180°

radians30° � 30 # p180

�p

6

The domain of the sine and cosinefunctions [6.6]

The domain of each of the sine andcosine functions is all angles, bothpositive and negative.

The domains are all real numbers.

The graph of y � sin x [6.6] The graph of is a periodic wave.One cycle is shown in the Example.

y � sin x

The period of the sine and cosinefunctions [6.6]

The period is the number of unitsrequired to complete one cycle of thegraph of a function.

The period is 360° for and .y � cos x

y � sin x

The graph of y � cos x [6.6] The graph of is a periodicwave. One cycle is shown in the Example.

y � cos x

Radian measure [6.7] Radian measure is used when the inputof a repeating function is better definedby real numbers than angles measured indegrees.

360° is equivalent to radians;the circumference of a unit circle.

2p

Converting from degree measure toradian measure [6.7]

To convert radian measure to degreemeasure, multiply the radian measure by

.180°pradians

� 120°

2p3

�2p3

# 180°p

�360°

3

The range of the sine and cosinefunctions [6.6]

The range of the sine and cosinefunctions is all values of N such that

.�1 � N � 1

The range is all real numbersfrom �1 to 1 inclusive.

Central angle [6.6] A central angle is an angle whose vertexis the center of a circle.

Problem 2b, Activity 6.6,page 697

06-100 ch 06 pp3 1/6/07 10:43 AM Page 744

745


The period of sine and cosine [6.8] For the trigonometric functions definedby or

, the period is 2pb .b 7 0y � a cos 1bx 2,y � a sin 1bx 2

The period of isunits.2p

2 � p

y � sin 12x 2

The displacement, or phase shift, ofthe graph of sine or cosine [6.9]

For the functions and, , the phase

shift is given by .�cb

b 7 0y � a cos 1bx � c 2y � a sin 1bx � c 2 The displacement of the function

is given by

. The shift is units

to the left.

p

6�

p2

3� �

p

6

y � 2 sin 13x �p2 2

The frequency of sine and cosine[6.8]

The frequency of trigonometricfunctions defined by or

, is the numberof cycles completed by the graphs overintervals of length The frequencyof trigonometric functions defined by

or , ,is b.

b 7 0y � a cos 1bx 2y � a sin 1bx 2

2p .

b 7 0y � a cos 1bx 2,y � a sin 1bx 2

The frequency of is2; two complete cycles occur in

units.2p

y � sin 12x 2

The displacement, or phase shift, ofthe graph of sine or cosine [6.9]

The displacement, or phase shift, of thegraph of , , isthe smallest movement (left or right)necessary for the graph of tomatch the graph of exactly. Similarly for the cosine.

y � a sin 1bx � c 2y � a sin bx

b 7 0y � a sin 1bx � c 2Examples 1 and 2, Activity 6.9page 726

The sine function can be used tomodel periodic wavelike behavior[6.10]

Using the TI-83/TI-84 Plus calculator setyour MODE to radian and use theSinReg option on the Stat Calc menu.

Problems 1–4, Activity 6.10pages 730–731

06-100 ch 06 pp3 1/6/07 10:43 AM Page 745

06-100 ch 06 pp3 1/6/07 10:43 AM Page 746

747

1. You walk 7 miles in a straight line 63° north of east.

a. Determine how far north you have traveled.

mi.

b. Determine how far east you have traveled.

mi.

2. Solve the following triangles.

a. b.

a. side c � 13, angle , angle

b. side , side , angle

c. d.

c. side , side , angle

d. side , side , angle

3. a. Given , determine and without using your calculator.

b. Given , determine , , and without using your calculator.

u � 30°

tan u �123

sin u �12

utan usin ucos u �232

tan u �68

cos u �810

tan ucos usin u �610

B � 30°c � 10a � 8.66

B � 45°c � 4.24b � 3

A

C B

560°

3

45°

A

C B

A � 60°b � 4a � 6.93

B � 22.6°A � 67.4°

A

C B

8

30°

A

C B

5

12

E � 7 cos 163° 2 � 3.18

N � 7 sin 163° 2 � 6.24

Gateway ReviewC h a p t e r 6

Answers to all Gateway exercises are included in the Selected Answers appendix.

06-100 ch 06 pp3 1/6/07 10:43 AM Page 747

c. Given , determine and without using your calculator.

4. You are taking your nephew to see the Empire State Building. When you are 100 feetaway from the building, you and your nephew look up to see the top. You are 6 feettall, your nephew is 3 feet tall, and the Empire State Building is 1414 feet high. Younotice that even though he is only half your height, your nephew does not have totilt his head any more than you do. Is your observation correct? Is the angle alwaysindependent of people’s heights? Explain.

No; there is a difference, but it is so small that it is difficult to see.

for me:

for my nephew:

5. In the diagram below, determine the lengths of a, b, c, and h to the nearest tenth.

a � 46.2

30 tan 57° � a

tan 57° �a

30

a

c

bh

13°

57°

30

u � tan �1

11411100 2 � 85.9461°

u � tan �1

11408100 2 � 85.9375°

cos u �5289

sin u �8289

c � 289

c 2 � 52 � 82 � 89

cos usin utan u �85

b � 10.7

b � 46.2 tan 13°

tan 13° �b

46.2

c � 55.1

c �30

cos 57°

cos 57° �30c

h � 47.4

h �46.2

cos 13°

cos 13° �46.2

h

748

06-100 ch 06 pp3 1/6/07 10:43 AM Page 748

6. In the following diagram, determine x and h. (Hint: See Exercises 2c and d.)

So,

x �400 ; 2480,000

4

x �400 ; 2160,000 � 320,000

4

x �400 ; 24002 � 412 2 140,000 2

212 2

2x 2 � 400x � 40,000 � 0

4x 2 � x

2 � 40,000 � 400x � x 2

12x22 � x 2 � 1200 � x22

The negative does not make sense,

thus, .

;

7. Using the following triangles, complete the table without using your calculator.

21

3√⎯

60°

30°1

1

45°

2√⎯

45°

h � 300 � 10023x � 100 � 10023

x �400 � 40023

4

x

h

45° 60°

30°

45°

200

2x =200 + X

120°

135° �1

150°

180° 0 �1 0

210°

225° 1

240°

270° �1 0 undef.

300°

315° �1

330°

360° 0 1 0

�123

232�

12

122�

122

�2312�

232

23�12�

232

�122

�122

123�232�

12

�123

�232

12

�122

122

�23�12

232

ANGLE uu SIN uu COS uu TAN uu

749

06-100 ch 06 pp3 1/6/07 10:43 AM Page 749

750

8. Determine the amplitude and period of the given functions, and then sketch theirgraphs. Use your graphing calculator to verify your results.

a.amplitude: 2

period:

2pp–2

3p—2

–1

–2

1

2

p x

y

2p

y � 2 sin x b.amplitude: 2

period:

2pp–2

3p—2

–1

–2

1

2

p x

y

2p

y � �2 sin x

c.amplitude: 1

period: p

pp–2

–1

1

x

y

y � cos 2x d.amplitude: 1

period: 1

0.5 1

–1

1

x

y

y � cos 2px

e.amplitude: 1

period: 4p

4p3p

2p

–1

1

p x

y

y � sin x2 f.

amplitude: 1

period: 4

1 2 43

–1

1

x

y

y � sin p x2

06-100 ch 06 pp3 1/6/07 10:43 AM Page 750

751

g.

amplitude: 1

period: 2p

2π

–1

1

π x

y

y � cos x h.

amplitude:

period: p

π–2

2–3

2–3

–

π x

y

23

y �23 cos 12x 2

i.amplitude: 1

period: 1

10.5

–1

1

x

y

y � sin 12px � 3p 2 j.amplitude: 3

period: 1

10.5–1

–2

–3

1

2

3

x

y

y � 3 sin 12px � 3p 2

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752

9. Match the given equation to one of the accompanying graphs. (Assume that Xscl 5 1and Yscl 5 1.)

a.Graph is vi.

y � �3 sin x b.Graph is ii.

y � 2 sin 1 p x2 �

p2 2

c.Graph is iv.

y � 2 sin 1 p x2 �

p2 2 d.

Graph is v.

y � �3 cos x

e.Graph is vii.

y � 2 cos 1 p x2 �

p2 2 f.

Graph is viii.

y � � cos 1px �p2 2

i. ii.

vii. viii.

v. vi.

iii. iv.

06-100 ch 06 pp3 1/6/07 10:44 AM Page 752

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the-eye.eu Books/Mathemati… · 655 CHAPTER 6 Introduction to the Trigonometric Functions ACTIVITY 6.1 The Leaning Tower of Pisa OBJECTIVES 1. Identify the sides and corresponding