chapter 14: trigonometry

T H E M E : Navigation

Early explorers relied on the stars and simple tools such as sextants and quadrants to find their way.

How did they do it? Long ago, people realized thatthe stars move in predictable patterns. By keepingcareful records and taking angle measurements,they discovered a way to pinpoint their locationon the Earth’s surface with a reasonable degreeof accuracy.

In the same way, modern navigators useinformation from satellites and guidance computersto find their way. Even automobiles are now equippedwith global positioning systems which use data fromsatellites to determine an automobile’s exact location in case ofan emergency. These advances are made possible by a branch of mathematicscalled trigonometry. Trigonometry, which means “triangle measurement,” isthe study of relationships among the sides and angles of a triangle.

• Commercial Fishers (page 633) Aside from fishing duties,commercial fishers pilot smallships or boats and must be ableto navigate to fishing areas. They use the stars as well aselectronic equipment to pinpoint their location.

610 mathmatters3.com/chapter_theme

Trigonometry

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Chapter 14 Trigonometry

Use the table for Questions 1–4.

1. On average, how many passengers arrive or depart from LAXeach day?

2. If the passenger traffic for both Denver and Los Angeles continueto change at the same rate, in what year would you expect Denverto have surpassed Los Angeles’ total passengers?

3. Of the airports shown on the table, which had the greatestdecrease in actual numbers of passengers from 2002 to 2003?

4. If the total number of passengers traveling through Dallas/Ft.Worth was 30,343,500 in 2000, what was the percent changebetween 2000 and 2003? Round to the nearest tenth of a percent.

CHAPTER INVESTIGATIONIn the Northern Hemisphere, the stars appear to move in a circularmotion around a single star named Polaris, commonly known asthe North Star. Explorers first navigated the globe using the star andan instrument called a quadrant.

Working TogetherBuild a quadrant using a photocopy of a protractor, heavycardboard, string, and a small weight. Use the quadrant to find theangles of elevation for several tall objects. Use trigonometricrelationships to find the height of the objects. Use the ChapterInvestigation icons to guide your group.

Data Activity: U.S. Airport Traffic

611

56,485

2009

Miami

�19.3%

U.S. Airport Traffic

AirportTotal Total

Percent changepassengers passengers2002–20032002 2003

Atlanta, Hartsfield (ATL) 37,070,492 38,228,500 3.1%Chicago, O’Hare (ORD) 28,356,224 30,797,513 8.6Los Angeles (LAX) 20,320,299 20,913,455 2.9Dallas/Ft. Worth (DFW) 24,072,162 24,502,273 1.8San Francisco (SFO) 12,250,289 12,227,636 �0.2Denver (DEN) 16,053,940 17,271,507 7.6Miami (MIA) 11,125,611 11,049,687 �0.7Newark (EWR) 13,113,997 13,087,544 �0.2Memphis (MEM) 4,537,659 4,504,679 �0.7

The skills on these two pages are ones you have already learned. Review theexamples and complete the exercises. For additional practice on these and moreprerequisite skills, see pages 654-661.

NAMING SIDES OF TRIANGLES

In the study of trigonometry, it is very important to be able to shift your focusand see the triangles from different points of view.

Examples Name the leg of �ABC that isadjacent to �B.

• Adjacent means “next to.” The adjacent side of an angle is never the hypotenuse (A�B�). Therefore, the adjacent side must be B�C�.

Name the side opposite �A.

• The side opposite an angle does not contain the vertex of the angle. Therefore, it must be B�C�.

Name the sides in each triangle.

1. The side opposite �S.2. The side adjacent to �S.3. The side adjacent to �R.4. The side opposite �R.

5. The side opposite �F.6. The side adjacent to �F.7. The side opposite �G.8. The side adjacent to �G.

9. The side opposite �M.10. The side opposite �L.11. The side adjacent to �L.12. The side adjacent to �M.


14CH

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612

14 Are You Ready?Refresh Your Math Skills for Chapter 14

A

B

COpposite

Adjacent

A

B

C

Opposite

Adjacent

R

ST

EF

G

L

MN

R�T�S�T�R�T�

S�T�

E�G�E�F�

E�F�E�G�

L�N�M�N�

L�N�M�N�

RATIONALIZING RADICALS

You will have to rationalize radicals much of the time while studyingtrigonometry.

Rationalize. Write in simplest radical form.

13. 14. ��31

20�� 15. 16. ��

435��

17. 18. ��356�� 19. 20. ��

584��

SPECIAL RIGHT TRIANGLES

• In a 30°-60°-90° triangle, the measure of the hypotenuse is two times that of theleg opposite the 30° angle. The measure of the other leg is �3� times that of the leg opposite the 30° angle.

• In a 45°-45°-90° triangle, the measure of the hypotenuse is �2� times themeasure of a leg of the triangle.

Find the unknown measures.

21. 22. 23.

24. 25. 26.

27. 28. 29. 9 cm

x cm

30°

15 cm

x cm

x cm5 cm

60°

x cm

18 cmx cm

60°

10 cmx cm

x cm

12 cm30°

x cm

x cm

2 cmx cm

x cm

8 cmx cm

x cm4 cm

30°

�27��7�

�96��24�

�18��5�

�8��3�

613Chapter 14 Are You Ready?

�2�

36�

�

2

�4�

55�

�

�6�

55�

�

�3�

510��

�3�

721��

�15�

�3�

23�

�

8 cm

4�2� cm �2� cm

9 cm5�2� cm4�3� cm

5�3� cm

�15

2�2�� cm

3�3� cm

Work with a partner.

Measure the segment lengths needed to calculate thefollowing ratios. Use a calculator to evaluate the ratios.

a. �AD

DE� b. c. �

BA

CB�

What conclusions can you draw from your results?

BUILD UNDERSTANDING

Precise measurement shows that the ratio of the lengths of two sides of a right triangle depends on the angles of the triangle—but not the lengths of the sides. This fact forms the foundation for the study of trigonometry. In this lesson, you will study the most important trigonometry ratios: the sine, the cosine, and the tangent.

In a right triangle, angle A is an acute angle. Then,

sine A �

cosine A �

tangent A �

Sine, cosine, and tangent are abbreviated sin, cos, and tan. Sin Ameans “the sine of �A.”

E x a m p l e 1

a. Find sin M.

b. Find cos M.

c. Find tan N.

Solutiona. sin M � b. cos M � c. tan N �

� �153� � �

11

23� � �

152�

� 2�15

�

opposite��adjacent

adjacent��hypotenuse

opposite��hypotenuse

length of leg opposite �A��length of leg adjacent to �A

length of leg adjacent to �A��

length of hypotenuse

length of leg opposite �A��


FG�AF

Chapter 14 Trigonometry614

14-1 Basic Trigonometric RatiosGoals ■ Identify trigonometric ratios in a right triangle.

■ Use trigonometric ratios to solve problems.

Applications Navigation, Construction, City Planning

A C

B

D

F

E G

35�

M P

N

12

513

Mental Math Tip

Use the memorydevice SOH CAH TOA(pronounced“sokatoah”) toremember thetrigonometric ratios.

SOH: Sine is OppositeoverHypotenuse.

CAH: Cosine isAdjacent overHypotenuse.

TOA: Tangent isOpposite overAdjacent.

In a right triangle with a 35° angle, the ratio of the length of the side oppositethe 35° angle to the length of the hypotenuse is approximately 0.57.

E x a m p l e 2

NAVIGATION A navigator at N sights a 37° angle between a buoy at B and a landmark at L. Find sin 37°.

SolutionYou can use the Pythagorean Theorem to find the length of the hypotenuse.

1802 � 2402 � h2

90,000 � h2

300 � h

sin 37° � �13

80

00

�

� 0.6

You can use your calculator to findtrigonometric ratios. Use the MODE key to setyour calculator to compute in the “degree”mode. Your calculator may already be set fordegree mode. Some calculators show the lettersDEG in the display window. To check yourcalculator’s mode, find sin 30°. The displayshould read 0.5. If your calculator displays�.988032, your calculator is in “radian” mode.

To find the angle that has a given trigonometricratio, use the inverse function on yourcalculator.

E x a m p l e 3

CALCULATOR An angle has a cosine of 0.55. Find the measure of the angle tothe nearest degree.

Solution

On a graphing calculator, press [cos-1] .55 .

(Some calculators use rather than .)

The display should read 56.6329. . .. To the nearest degree, an angle with a cosine of 0.55 measures 57°.

cos–1cosINV

) 2nd

Lesson 14-1 Basic Trigonometric Ratios 615

B

L

N37�

240 m

180 mh

CheckUnderstanding

In the figure at the topof page 614, which leg isopposite �A in �ADE?Which leg is adjacent to�AFG in �AFG?

DE; FG

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TRY THESE EXERCISES

Use the figure at the right to find each ratio. Expressanswers in lowest terms.

1. tan K 2. cos H 3. sin K

4. CONSTRUCTION A 17-ft wire is attached near the topof a wall. The wire is then anchored to the ground 15 ftfrom the base of the wall. Find tan A to the nearesthundredth.

CALCULATOR Use a calculator to find each ratio. Roundto the nearest ten-thousandth.

5. sin 22° 6. cos 81° 7. tan 52°

8. cos 40° 9. tan 12° 10. sin 64°

PRACTICE EXERCISES • For Extra Practice, see page 707.

Use the figure at the right to find each ratio.

11. tan G 12. sin F

13. cos F 14. sin G

15. tan F 16. cos G

17. In �ABC, �C is a right angle, AB � 29, and AC � 21.Find sin A and tan B to the nearest hundredth.

18. In right triangle RST, �T is the right angle and tan R � �

490�. Write sin R and tan S as ratios.

19. NAVIGATION An airline navigator measures a 16° angle between thehorizontal and an ocean liner. Find tan A to the nearest hundredth.

Find angles to the nearest tenth of a degree.

20. CITY PLANNING Filbert Street and 22nd Street in San Francisco are thenation’s steepest streets. Each rises 1 ft for every 3.17 ft of horizontaldistance. What angle do these streets form with the horizontal?

21. NAVIGATION From a boat at sea, the distance to the top of a 2325-ft cliff atthe water’s edge is 4370 ft. What angle does a line make with the horizontalfrom the boat to the top of the cliff?

22. A kite at the end of 545 ft of string is 130 ft above the ground. What angledoes the kite string make with the ground?

24 mi

25 mi

A


F H

G

12

915

15A

17

J H36

1539

K

�152� �

1123� �

1123�

0.53

0.3746

0.7660

0.1564

0.2126

1.2799

0.8988

�43

�

�45

�

�34

�

�35

�

�45

�

�35

�

�491�; �49

0�

about 17.5°

about 32.1°

about 13.8°

0.29

0.69; 1.05

Lesson 14-1 Basic Trigonometric Ratios 617

NAVIGATION The table at the right showsmeasurements taken at six East Coast lighthouses.Use the table for Exercises 23–24.

23. The navigator of a ship standing on its bow360 ft from the McClellanville lighthousemeasures a 23° angle to the top of thelighthouse. Write cos 23° as a ratio.

24. A boat pilot standing on the bow 1120 ft froma lighthouse measures an 8° angle to the top of the lighthouse. If cos 8° � �

11

11

23

�, where is the lighthouse located?

25. WRITING MATH Suppose the top of a lighthouse measured 10°from your position offshore. If you knew the height of thelighthouse, how could you find your distance from shore?

26. CHAPTER INVESTIGATION Your task is to have each member ofyour group build a homemade quadrant and use them to find theheights of various objects. Gather the materials you will need: oneplastic straw or unsharpened pencil per person, string, a weightsuch as a metal washer or bolt, and tape. Make a photocopy of aprotractor or draw a protractor on heavy cardstock. If you chooseto draw your own protractor, make and label markings for everyten degrees.

EXTENDED PRACTICE EXERCISES

Find the exact value of each ratio.

27. tan 45° 28. cos 30°

29. cos 60°

30. WRITING MATH True or false: In a right triangle, the sine of one acute angle equals the cosine of the other. Explain.

MIXED REVIEW EXERCISES

Write an equation for each circle. (Lesson 13-1)

31. radius 5 32. radius 8 33. radius 3.5center (0, 0) center (4, 3) center (5, 1)

34. radius 10 35. radius 7 36. radius 4.7center (�2, 0) center (3, �2) center (�2, �2)

37. radius 6 38. radius 2 39. radius 7.5center (�3, 5) center (2, �8) center (�4, 3)

A bag contains 6 red marbles, 5 green marbles, 8 blue marbles, and 1 whitemarble. Marbles are taken from the bag at random one at a time and notreplaced. Find each probability. (Lesson 9-4)

40. P(red, then blue) 41. P(green, then white)

42. P(green, then blue, then red) 43. P(white, then blue, then blue)

Location ofLighthouse

Height(ft)

Annisquam, MA 41Cape May, NJ 170Fenwick Island, DE 87McClellanville, SC 150Millbridge, ME 123Scituate, MA 25

A C

B

b

ac

�1123�

Check students’ work.

McClellanville

Divide the height of the lighthouse by the ratio of height to distance (or tan 10°).

1 ��23�

�

�1925�

�527�

�716�

�8

755�

�12

�

True. In the figure, sin A � �ac

� � cos B.

x2 � y2 � 25

(x � 4)2 � (y � 3)2 � 64 (x � 5)2 � (y � 1)2 � 12.25

(x � 2)2 � ( y � 2)2 � 22.09

(x � 4)2 � (y � 3)2 � 56.25(x � 2)2 � (y � 8)2 � 4

(x � 3)2 � (y � 2)2 � 49

(x � 3)2 � (y � 5)2 � 36

(x � 2)2 � y2 � 100

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Work with a partner. Use the figures below to answer these questions.

a. How do you know the triangles are similar?

b. Write a proportion you could solve to find x.

c. Solve the proportion for x.

d. Explain how you could solve the equation sin 30° � �1x2� for x.

e. Solve for x: sin 40° � �1x6�. Explain how you solved the equation.

BUILD UNDERSTANDING

You can find the measures of missing parts of a right triangle.

If you know the measure of one acute angle, you can find the measure of theother by subtracting the measure of the known angle from 90°.

If you know the lengths of two sides, you can use the Pythagorean Theorem tofind the length of the third side.

If you know the measure of an angle and the length of a side, you can usetrigonometric ratios and the first two rules to find the other parts of the triangle.

E x a m p l e 1

Find the following in �PQR.

a. PR to the nearest tenth

b. m�P

c. RQ to the nearest tenth


14-2 Solve RightTrianglesGoals ■ Find the lengths of sides and the measures of angles

in right triangles.

Applications Surveying, Navigation, Safety

12

x12

30� 30�

R Q

P

42

27�

They contain two pairs of correspondingcongruent angles.

�12

� � �1x2�

x � 6Write sin 30° as �

12

�, then cross-multiply to find x � 6.

sin 40° � 0.6428, x � 16(0.6428) � 10.2848

Solutiona. To decide which trigonometric ratio relates P�Q�, P�R�, and �Q,

think: P�Q� is the hypotenuse and P�R� is opposite �Q. The ratio thatrelates the hypotenuse and the leg opposite an angle is the sine.

sin Q � �PP

QR�

sin 27° � �P4

R2�

0.4540 � �P4

R2� calculator approximation of sin 27°

PR � 42(0.4540) � 19.1

b. m�P � 90° � 27° � 63°

c. Use the Pythagorean Theorem to find RQ.

PR2 � RQ2 � PQ2

19.12 � RQ2 � 422

364.81 � RQ2 � 1764

RQ2 � 1399.19

RQ � 37.4

Finding the measures of all parts of a right triangle iscalled solving a right triangle.

Trigonometry problems often involve angles ofdepression and elevation.

An angle of depression is formed by a horizontal line anda line slanting downward.

An angle of elevation is formed by a horizontal line and aline slanting upward.

E x a m p l e 2

SURVEYING A surveyor is standing 550 ft from the base of a redwood tree inHumboldt County, CA. The tree is 362 ft tall. Find the angle of elevation of the topof the tree from the spot where the surveyor is standing.

SolutionThe angle of elevation is �A, formed by the horizontalline of the ground and the line slanting to the top of the tree. BC is opposite �A, and AC is adjacent to �A. The trigonometric ratio relating opposite and adjacent isthe tangent.

tan �A � �35

65

20

�

tan�1 ��35

65

20

�� 33.4°

The angle of elevation is approximately 33.4°.

Lesson 14-2 Solve Right Triangles 619

Angle of depression

Angle of elevation

550 ft

362 ft

A C

B

Math: Who,Where, When

Hipparchus, a Greekmathematician bornabout 160 B.C., isgenerally considered tobe the creator oftrigonometry. He was thefirst person to draw up atable of values for thesine, cosine, and tangentratios.


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TRY THESE EXERCISES

Find the following in �JKL.

1. LK to the nearest tenth

2. JK to the nearest tenth

3. m�J

Find the following in �CRL.

4. LR to the nearest tenth

5. m�C to the nearest degree

6. m�R to the nearest degree

7. NAVIGATION A ship’s sonar detects a submarine 880 ft below a point on the ocean’s surface 1450 ft dead ahead of the ship. To the nearest degree, find the angle of depression to the submarine.

8. SAFETY Safety experts recommend that a ladder be placed at an angle of about 75° to the ground. Mr. Reese is using a 15-ftladder. How far from the base of the wall should he place the footof the ladder? Round the distance to the nearest tenth of a foot.


Round each length to the nearest tenth and each angle to the nearest degree.

CALCULATOR Use a calculator to find the following in �ABC.

9. BC

10. m�B

Find the following in �XYZ.

11. YZ

12. m�Z

13. m�X

14. Find the angle of elevation to the top of the 1250-ft Empire State Buildingfrom a point 850 ft from the base.

15. The two legs of a right triangle measure 23.5 and 27.9. Solve the triangle.

16. In a right triangle, the leg adjacent to a 77° angle has a length of 87. Solve thetriangle.

17. AIR TRAFFIC CONTROL From an airport runway, the angle of elevation toan approaching plane is 12.8°. If the plane’s altitude is 2400 ft, how far is theplane from the runway?


J

K

L

56�

32

L R

8.3

C

14.5

A C

B

4

5

37�

Y

Z

X

18.6

15.2

21.6

38.6

34°

11.9

55°

35°

31°

3.9 ft

3.0

53°

10.7

55°

35°

56°

hypotenuse: 36.5; angles: 40°, 50°, 90°

angles: 13°, 90°; legs: 376.8; hypotenuse: 386.8

10,832.8 ft

18. NAVIGATION From the top of a cliff, the angle of depression of aship at sea is 8.8°. If the direct-line distance from the clifftop to theship is 2.3 mi, how high is the cliff?

19. BOATING The foot of a right-triangular sail is 64 in. long. The angleat the top of the sail measures 23°. Find the length of the luff of thesail.

20. FOREST MANAGEMENT A ranger in a fire tower spots a fire at an angle of depression of 4°. The tower is 36 m tall. How far from the base of the tower is the fire?

21. A silo casts a shadow 42 ft long. The angle of elevation from the sun tothe ground is 38°. How tall is the silo?

22. DATA FILE Use the data on highest and lowest continental altitudes onpage 647. A ship’s navigator at sea level in Cook’s Inlet, Alaska, sights thesummit of Mount McKinley at a 6.8° angle of elevation. How many miles is itfrom the ship to the summit of the mountain?

23. FLIGHT A helicopter ascends 150 ft vertically, then flies horizontally 420 ft.Find the angle of elevation to the helicopter as seen by an observer at thetakeoff point.

24. COMMUNICATIONS Orlando and Ryan are taking measurements related tothe installation of a TV tower. Orlando measures a 62° angle of elevation tothe top of the 950-ft TV tower. Find the angle of elevation for Ryan, standing80 ft farther from the tower than Orlando.

25. CHAPTER INVESTIGATION Assemble the quadrant in the followingmanner: Tape the protractor image to the straw (or pencil), aligning the flatedge of the protractor with the length of the straw. Tie the weight to one endof a 8-in. length of string. Secure the other end of the string to the straw orpencil at the center point of the protractor base. The string should be able tomove freely. As you sight an object through the straw, the string will indicatethe angle of ascent or descent on the protractor scale.


Can you solve a right triangle from the given information? Answer yes, no, orsometimes.

26. two sides 27. three angles

28. one side 29. one leg and one angle

30. A pike is directly beneath a trout in a lake. The direct-line distance from anangler to the trout is 35 ft. The angle of depression to the trout is 20°. Theangler’s direct-line distance to the pike is 42 ft. The angle of depression to thepike is 24°. How far below the trout is the pike?


Find the focus and directrix of each equation. (Lesson 13-2)

31. x 2 � 4y 32. x 2 � �5y 33. x 2 � 8y

34. x 2 � 6y � 0 35. x 2 � 10y � 0 36. x 2 � 18y

Lesson 14-2 Solve Right Triangles 621

foot

luff

0.4 mi

150.8 in.

514.8 m

32.8 ft

32.5 mi

20°

58°

yes

no

no

5.1 ft

(0, 1); y � �1

34. �0, �32

��; y � ��32

� 35. �0, �52

��; y � ��52

� 36. �0, �92

��; y � ��92

�

(0, 2); y � �2�0, ��54

��; y � �54

�

Sometimes; the angle must be acute.




PRACTICE LESSON 14-11. For what type of triangle can you apply the trigonometric ratios?

Determine if each statement is true or false.

2. The sine of an acute angle is calculated by dividing the length of the oppositeleg by the length of the hypotenuse.

3. The tangent of an acute angle is calculated by dividing the length of thehypotenuse by the length of the adjacent leg.

4. The cosine of an acute angle is calculated by dividing the length of theadjacent leg by the length of the hypotenuse.

Use the figure at the right to find each ratio. Express answersin lowest terms.

5. tan A 6. sin C 7. cos C

8. tan C 9. sin A 10. cos A

11. A 120-ft flagpole casts a shadow of 90 ft. Write sin D as a ratio.

12. In �ABC, �C is a right angle, AB � 18 and AC � 12. Findsin A and tan B to the nearest hundredth.

13. In �RST, �T is a right angle, RS � 10, and ST � 5. Find cos Sand tan R to the nearest hundredth.

PRACTICE LESSON 14-214. Explain how you can use the Pythagorean Theorem to find the length of a

side of a right triangle if you know the lengths of two sides.

15. Explain how to find the measure of an unknown acute angle in a righttriangle.

Find the following in �NOP.

16. NO to the nearest tenth

17. m�N to the nearest degree

18. m�P to the nearest degree

Find the following in �RST.

19. RT to the nearest tenth

20. RS to the nearest tenth

21. m�T

22. To the nearest tenth, find the angle of elevation to the top of the 1454-ftSears Tower from a point 750 ft from the base.


N

P

O

15.89.2

T

R S

48

30°

A

C

B

10 6

8

90 ft

120 ft

D

right triangle

true

false

true

�45

� �35

�

�45

��35

�

�34

�

�43

�

�45

�

12.8

36°

54°

24.0

41.6

60°

about 62.7°

sin A � 0.75; tan B � 0.89

cos S � 0.50; tan R � 0.58

For 14–15, see additional answers.

Review and Practice Your Skills

PRACTICE LESSON 14-1–LESSON 14-2In the figure at the right, use the Pythagorean Theorem to find the missingmeasure to the nearest hundredth. Then find each ratio to the nearesthundredth. (Lesson 14-1)

23. sin L 24. cos L

25. cos M 26. tan L

27. sin M 28. tan M

29. In �DEF, �F is a right angle, DE � 30, and EF � 24. Findsin D and tan E to the nearest hundredth.

30. In �LMN, �N is a right angle, LM � 84, MN � 62. Find cos L and tan M tothe nearest hundredth.

Find the following in �ABC. (Lesson 14-2)

31. m�C

32. BC to the nearest tenth

33. AB to the nearest tenth

34. Find the angle of elevation to the top of a 150-ft flagpole from a point 45 ftfrom the base of the pole.

35. The direct-line distance from the top of the slope to the ski lodge is 2500 ft.The top of the slope is 1050 ft above the level of the lodge. What is the angleof depression from the top of the slope to the lodge?

Mid-Chapter QuizSolve.

1. A ladder on a 10-ft tall fire truck is 75 ft long. If it makes a 45° angle with abuilding, what is the greatest height the ladder can reach up the side of thebuilding? (Lesson 14-1)

2. When viewed from a horizontal distance of 32 ft, the top of a flagpole can beseen at an angle of 39°. What is the height of the flagpole? (Lesson 14-1)

3. Right triangle ABC has hypotenuse A�C�. Right triangle DEF with hypotenuse D�F�is similar to ABC, and angle D corresponds to angle A. If D�E� measures 15 andE�F� measures 20, what is the value of tan A? (Lesson 14-1)

4. An office worker on the fourteenth floor of a building sights a friend on thestreet. The angle of depression is 35°, and the fourteenth floor is 135 ft in theair. How far is the friend from the building? (Lesson 14-2)

5. The pilot of a plane flying east sights another plane ahead of him at an angle of elevation of 18°. The line of sight distance between the planes is 1850 m. At how much greater altitude is the lead plane than the trailing one?(Lesson 14-2)

Chapter 14 Review and Practice Your Skills 623

A

C

B

36

24°

M

L N

14 8

0.57

0.57

0.82

0.82

0.70

1.44

sin D � 0.8; tan E � 0.75

cos L � 0.67; tan M � 0.91

66°

14.6

32.9

73.3°

about 24.8

63 ft

about 26 ft

�43�

about 192.8 ft

about 571.7 m

Work with a partner. You will need a calculator.

1. Choose several angles with measures in each of the given ranges below, and find the sine and cosine of each.

a. 90°–180° b. 180°–270° c. 270°–360°

2. Draw a grid like the one shown below. Graph your results for parts a, b, and c.

3. Study each range of angles. Describe any patterns you observe in the signs (� and �) of the sine ratio and the cosine ratio.

BUILD UNDERSTANDING

Until now, you have dealt only with acute angles in your work with trigonometricratios. In this lesson, you will learn to find trigonometric ratios of angles withmeasures greater than or equal to 90°. To solve these problems, you will need toapply the relationships that hold in 30°-60°-90° and 45°-45°-90° right triangles.

E x a m p l e 1

Find sin 240°.

SolutionSketch the angle on a coordinate plane. Use thepositive x-axis as the initial side, and the ray resultingfrom a 240° counterclockwise rotation of the positivex-axis as the terminal side. The reference angle is the acute angle formed by the x-axis and the terminal side.

1

1

45�

45�

��2

��3

12

30�

60�

90° 180° 270° 360°

y

x

sin x

cos x�1

1


14-3 Graph the Sine FunctionGoals ■ Solve problems using trigonometry.

Applications Communications, Population, Flight

y

x

240�

�120�

initial side

terminalside

referenceangle

a: sine �, cosine �; b: sine �, cosine �; c: sine �, cosine �

The reference angle measures 240° � 180° � 60°. Complete atriangle with the terminal side ashypotenuse by drawing a segmentperpendicular to the x-axis. Thetriangle is a 30°-60°-90° righttriangle. In this example, thetriangle is in the third quadrantbecause the legs were drawn bymoving in negative directions fromthe x- and y-axes. The length of theterminal side is always consideredto be positive.

To find sin 240°, find the sine of the reference angle.

sin 240° � �

E x a m p l e 2

Find sin 495°.

SolutionTo form an angle of 495°, the initial sidemust complete a 360° rotation, thencontinue an additional 135°. The referenceangle measures 180° � 135° � 45°. Thetriangle formed is a 45°-45°-90° righttriangle. The leg adjacent to the 45° anglemeasures �1 relative to the x-axis. The legopposite the angle measures �1 relative tothe y-axis. The length of the terminal sideis always positive.

sin 495° � �

E x a m p l e 3

COMMUNICATIONS A researcher is developing new technology to assist shipsat sea to send urgent communications. Ships will use various tones and patternsto send messages. The electrical impulses produced by the tones are modeled bysine curves. Graph the sine curve y � sin x for 0° � x � 360°.

SolutionUse your calculator to make a table of ordered pairs.

Graph the points using sin x as the y-coordinate. Draw a smooth curve throughthe points. The graph is called a sine curve.

�2��

21

��2�

��3��

2opposite

��hypotenuse

Lesson 14-3 Graph the Sine Function 625

y

x�1

2

60�

30��3

y

x�1

145� ��2

45�

495� � 360� � 135�

TechnologyNote

Use your calculator tocheck trigonometricratios that you find usingreference angles. InExample 1,

��

23�

� � �0.8660.

Check using the key:

�0.8660254

ENTER) 240SIN

SIN



GRAPHING If you have a graphing calculator, you can easily graph sine curves.To graph y � sin x, make sure your calculator is in degree mode. To show therange of x-values, set the Xmin at 0 and the Xmax at 360. Use �1.5 (Ymin) and1.5 (Ymax) as the range of y-values. In the graph above, the x-scale is set at 45and the y-scale is at 1. You are ready to graph the function.

TRY THESE EXERCISES

Find each ratio by drawing a reference angle.

1. sin 135° 2. sin 300° 3. sin 210°

4. sin 405° 5. sin 660° 6. sin 855°

7. POPULATION A sine curve models the population growth for wildlife in awooded area. Graph y � sin x for 360° � x � 540°.

8. WRITING MATH Explain how you would find the reference angle and drawa right triangle for a 1050° angle.



9. sin 225° 10. sin 330° 11. sin 120° 12. sin 150°

13. sin 315° 14. sin 480° 15. sin 585° 16. sin 690°

17. sin 675° 18. sin 930° 19. sin 765° 20. sin 1200°

21. GRAPHING Use a graphing calculator to graph y � sin x for 360° � x � 720°.Use �1.5 and 1.5 as your range of y-values.


22. cos 120° 23. tan 225° 24. cos 330°

25. tan 240° 26. tan 660° 27. cos 765°

28. sin (�60°) 29. sin (�45°) 30. sin (�30°)

31. cos (�315°) 32. tan (�840°) 33. cos (�1230°)

34. Graph y � sin x for �360° � x � 0°.

35. Graph y � cos x for 0° � x � 360°.

36. WRITING MATH Compare and contrast the sine curve and the cosine curve.

Solve for values of x with 0° X x X 360°.

37. sin x � 0 38. cos x � �1 39. sin x � 1 40. cos x � 0

626

1

�1

90 180 270 360

y

x

y � sin x


��22�

�

��22�

�

��

23�

�

��

23�

�

��12

�

��22�

�

See additional answers.

This initial side must complete two 360° rotations, thencontinue an additional 330°. The reference angle measures 360° � 330° � 30°. Draw aperpendicular to the x-axis to form a 30°-60°-90° right triangle. The leg adjacent to the 30° anglemeasures �3�. The leg opposite measures �1. The hypotenuse measures 2.

��22�

�

��22�

�

��22�

�

��12

�

��23�

�

��12

�

��23�

�

��22�

�

��22�

�

�12

�

��12

�

��23�

�

��23�

��12

�

�3�

��23�

�

��22�

�

1

��3�

��22�

�

�3�

��22�

�

��12

�

��3��

2




180° 90° 90°, 270°


0°, 180°, 360°

41. FLIGHT Use a graph of the equation y � sin x to estimate the value of sin68°. Then use the value to find the length of a kite string pitched at a 68°angle to the ground if the kite is 450 ft above the ground.

42. CHAPTER INVESTIGATION Work in small groups to determine the height(or depth) of five objects on your school grounds or nearby community.

Follow these steps:

a. Stand at a particular point.

b. Measure the distance from the point to the base of the object.

c. Take an angle measure from the point using the quadrant.

d. Use your knowledge of trigonometry to determine the height of theobject. Remember to take into account the distance from your eye levelto the ground.

Share the results of your activity with the class. If more than one group measured the same object, compare measurements. Discuss how to account for any discrepancies.


43. a. Describe a method involving the graphs of y � sin x and y � cos x thatyou could use to solve the equation sin x � cos x.

b. Use your method to solve the equation for 0° � x � 360°.


44. sin x � ��12

�

45. cos x ��

46. tan x � �1

47. tan x � �3�

48. The trademark on a 26-in. radiusbicycle tire is 25.5 in. from the centerof the wheel. The spoke touching thetrademark is horizontal. Find theheight of the trademark above theground after the wheel turns throughan angle of 495°.


Find the equation of each ellipse. (Lesson 13-4)

49. foci: (2, 0) and (�2, 0) 50. foci: (2, 0) and (�2, 0)x-intercepts: (4, 0) and (�4, 0) x-intercepts: (3, 0) and (�3, 0)



�3��

2

Lesson 14-3 Graph the Sine Function 627

� 485 ft



x � 45° and x � 225°

210°, 330°

150°, 210°

135°, 315°

60°, 240°

about 44.0 in.

3x2 � 4y2 � 48

5x2 � 9y2 � 405

32x2 � 81y2 � 2592 3x2 � 4y2 � 192

5x2 � 9y2 � 45

11x2 � 36y2 � 396


trademark


Work with a partner to answer the following questions. You will need agraphing calculator.

1. Graph each equation on a graphing calculator. How does the coefficient of x 2

affect the shape of the graph?

a. y � x 2 b. y � 2x 2 c. y � 3x 2

2. Graph each equation. How does the constant in parentheses affect theposition of the graph?

a. y � x 2 b. y � (x � 1)2 c. y � (x � 2)2

BUILD UNDERSTANDING

Recall the sine curve you studied in the last lesson.

Notice that the shape of the curve repeats itself in every 360° interval along the x-axis. If you were to pick up a 360° section of the curve, you wouldfind it congruent to the curve in each adjacent 360° section. Functions with repeating patterns like this are called periodic functions. The periodof the function is the length of one complete cycle of the function. The period of the graph of y � sin x is 360°.

E x a m p l e 1

COMMUNICATIONS A tone transmitted to a ship at sea produces a sound wave with the equation y � sin 2x. State the period.


14-4 Experiment withthe Sine FunctionGoals ■ Determine the period, amplitude and position of sine

curves.

Applications Communications, Music, Medicine

1

�1

y

x

y � sin x



SolutionThe effect of the coefficient 2 in the equation is to compress the sine curve horizontally. The period of y � sin 2x is 180°, half the period of y � sin x.

Example 1 suggests the following rule.

The period of the graph of y � sin nx is �36n0°�.

Look again at the graph of y � sin x on the opposite page. Notice that the maximum value of y is 1 and the minimum value of y is �1.The amplitude of a periodic function is half the difference between its maximum and minimum y-values. The amplitude of y � sin x is �

12

�(1 � [�1]) � 1. Amplitude is a measure of the height and depth of a curve.

E x a m p l e 2

Graph y � 2 sin x. State the amplitude.

Solution

The graph of y � 2 sin x is twice as tall and twice as deep as the graph of y � sin x. The amplitude is �

12

�(2 � [�2]) � 2.

Notice that, in Example 2, the amplitude is the same as the coefficient in theequation y � 2 sin x. This suggests the following rule. The amplitude of the graphof y � n sin x is n.

GRAPHING Graph y � 2 sin x using a graphing calculator to check your work.You may need to change the values in the display window in order to see theentire curve. Set the y-maximum equal to or greater than the amplitude (n), andthe y-minimum equal to or less than �n.

E x a m p l e 3

Graph y � sin x � 1. Describe the position of the graph.

Solution

The graph of y � sin x � 1 is the graph of y � sin x raised 1 unit above its normal position.

Example 3 suggests the following rule.

The graph of y � sin x � n is the graph of y � sin x raised or lowered n units.

Lesson 14-4 Experiment with the Sine Function 629

y

x180 360�180

�1

1

�360

y � sin 2x

2

1

�2

y

�180 180 x

y � 2 sin x

2

1

�1

�2

180 360

y

x

y � sin x � 1

�180�360

Period 90°; amplitude 3;the graph is the graph ofy � 3 sin 4x lowered 2units.


CheckUnderstanding

Without drawing agraph, find the periodand amplitude of thegraph of y � 3 sin 4x � 2and describe the positionof the graph.


TRY THESE EXERCISES

1. Graph y � sin 3x. State the period.

2. Graph y � 4 sin x. State the amplitude.

3. Graph y � sin x � 1. Describe the position of the graph.

State the period and amplitude of the graph of each equation and describe theposition of the graph.

4. y � 3 sin x � 2 5. y � 2 sin 2x � 5

6. YOU MAKE THE CALL To graph y � sin x � 3, Cynthia enters y � sin (x � 3) on her graphing calculator. Has Cynthia made a mistake?Explain your thinking.


7. Graph y � sin 4x. State the period.

8. Graph y � 0.5 sin x. State the amplitude.

9. Graph y � sin x � 3. Describe the position of the graph.


10. y � 2 sin x � 1 11. y � 4 sin 3x � 2.5

Tell if the function is periodic. If it is, state the period.

12. 13.

14. 15.

16. MUSIC A note played on a musical instrument produces a sound wave withthe equation y � 3 sin 4x � 3. State the period and amplitude, and describethe position of the graph.

17. MEDICINE An animal’s heart rate can be modeled by a sine curve that has aperiod of 540° and an amplitude of �

32

�. Write the equation.

y

x2�2�

y

x2 4 6 8�2�4�6�8

y

x

5�5�10 10 15 20�15�20

y

x2�2�4 4 6 8�6�8


For 1–3, see additional answers for graphs.

period � 120°

The graph is the graph of y � sin x lowered 1 unit from the origin.

The period is 90°.

The amplitude is 0.5.

The graph is the graph of y � sin x raised 3 units.


For 7–9, see additional answers for graphs.

Yes. The expression sin x � 3 is the sum of sin x and 3. In the expression sin (x � 3), the sine is found for the sum of x and 3. The graphs of the two expressions are not the same.

amplitude � 4

no yes; 5

noyes; 6

period: 90°; amplitude: 3; graph is the graph ofy � 3 sin 4x raised 3 units.

Sample answer:

y � �32

� sin �23

�x


Find the period of the graph of each equation.

18. y � sin �12

�x 19. y � sin �35

�x 20. y � sin �172�

21. Find an equation of a graph involving the sine function that has a period of630° and an amplitude of 8.

State an equation for a sine function with the graph shown.

22. 23.


Graph the equation.

24. y � 2 cos x 25. y � cos 2x 26. y � cos x � 1

27. WRITING MATH Write rules you can use to find the period, amplitude, andposition of a graph involving the cosine function.

Make a table of ordered pairs and graph the equation.

28. y � �sin x 29. y � sin (x � 90°)

30. WILDLIFE MANAGEMENT The equation l � 50,000 � 48,000 sin 90t approximates the number of lemmings on an arctic island on January 1 of a year t years after January 1, 1980.

a. Find the maximum number of lemmings.

b. What year was the maximum first reached?

c. Find the minimum number of lemmings.

d. What year was the minimum first reached?


Solve each variation. (Lesson 13-5)

31. If y varies directly as x, and y � 9 when x � 6, find y when x � 27.



Factor the following trinomials. (Lesson 11-7)

34. x 2 � 3x � 10 35. 2x 2 � 5x � 12 36. x 2 � 2x � 35

37. 4x 2 � 8x � 5 38. x 2 � 6x � 9 39. 3x 2 � 3x � 6

40. 3x 2 � 14x � 5 41. x 2 � 64 42. 2x 2 � 6x � 56

�1

�2

y

x135 18090454

3

2

1

y

x1080 1440720360

Lesson 14-4 Experiment with the Sine Function 631

720° 600° 210°

Sample answer: y � 8 sin �47

�x

y � �12

� sin 8x � 1y � 2 sin �14

�x � 2




98,000

1981

2000

1983

40.5

120

133

(x � 5)(x � 2) (2x � 3)(x � 4)(x � 7)(x � 5)

3(x � 2)(x � 1)

2(x � 7)(x � 4)

(x � 3)(x � 3)

(x � 8)(x � 8)

(2x � 1)(2x � 5)

(3x � 1)(x � 5)



PRACTICE LESSON 14-3Determine if each statement is true or false.

1. The initial side of an angle is always on the positive x-axis.

2. The terminal side of an angle is found in the second quadrant.

3. The reference angle is the acute angle formed by the y-axis and the terminalside of the angle.


4. sin 270° 5. sin 120° 6. sin 150°

7. sin 300° 8. sin 315° 9. sin 585°


10. tan 210° 11. cos 240° 12. tan 135°

13. sin 330° 14. tan 300° 15. cos 225°


16. sin x � �1 17. cos x � �12

� 18. sin x � ��12

�

19. cos x � ��12

� 20. sin x � 1 21. cos x � 1

PRACTICE LESSON 14-4Define each term.

22. period 23. amplitude

24. Use the graph of y � sin 4x to state the period.

25. Use the graph of y � 2 sin x to state the amplitude.

Find the period of the graph of each equation.

26. y � 2 sin 5x 27. y � sin 3x � 2 28. y � 3 sin x � 4

29. y � 2 sin 10x 30. y � sin 6x � 1 31. y � 5 sin x � 2

32. y � 3 sin 4x � 1 33. y � sin 40x � 3 34. y � 2 sin 5x � 1


35. y � 2 sin x 36. y � sin 4x 37. y � 2 sin 3x � 1

38. y � 3 sin 6x � 3 39. y � 5 sin 2x � 2 40. y � 2 sin 5x � 1


true

false

false

�1

��

23�

�

��23�

�

��

22�

�

�12

�

��

22�

�

�1

��

22�

�

330°, 210°

0°, 360°

60°, 300°

��12

�

��3�

��33�

�

��12

�

270°

120°, 240° 90°

The length of one completecycle of the function.

Half the difference betweenthe maximum and minimumy-values.

90°

2

72° 120° 360°

36°

90°

60°

9°

360°

72°

120°, 2, shifted 1 unit up

72°, 2, shifted 1 unit down180°, 5, shifted 2 units up

90°, 1, centered on x-axis360°, 2, centered on x-axis

60°, 3, shifted 3 units down

Review and Practice Your Skills

Workplace Knowhow

PRACTICE LESSON 14-1–LESSON 14-4Use the figure at the right to find each ratio. (Lesson 14-1)

41. sin A 42. cos B

43. sin B 44. tan A

45. tan B 46. cos A

47. To the nearest tenth, find the angle of elevation to the topof a 90-ft tree from a point 40 ft from the base of the tree.(Lesson 14-2)

Find each ratio by drawing a reference angle. (Lesson 14-3)

48. sin 30° 49. sin 135° 50. sin 210°

51. cos 270° 52. tan 225° 53. cos 45°

Solve for values of x with 0° X x X 360°. (Lesson 14-3)

54. sin x � 55. sin x � 56. sin x �

57. cos x � 58. tan x � �1 59. cos x ��2��

2�3��

2

�2��

2�3��

2��3��

2

Chapter 14 Review and Practice Your Skills 633

Commercial fishers are captains, deckhands, or boatswains (supervisor of thedeckhands). Aside from fishing duties, the fishers aboard a fishing boat must

be able to navigate to the fishing areas. Today this is mainly accomplishedthrough the use of electronic equipment that pinpoints the ship’s position on thesurface of the Earth according to man-made satellites orbiting the planet. Beforethese devices were invented, sailors navigated according to the stars.

To avoid rocks near a shoreline, an experienced fisher uses the stars to knowwhen to turn and make an arc to shore. When under the correct star, the fisher is2.5 mi from shore. The arc begins there and ends 2.5 mi from the shore pointwhere the arc began. His route is a quarter of a circle with a radius of 2.5 mi.

1. How many miles does the fisher travel in his arc?

2. On a coordinate plane,name the x, y coordinates of the beginning and endingpoints of the arc.

3. What is the equation of the whole circle of which the fisher traveled onequarter?

A

B

C

857

6

�7�85

85�� 7�85

85��

�76

�

�6�85

85��

�6�85

85��

�67

�

66.0°

�12

�

0

��22�

�

1

��12

�

��22�

�

45°, 315°

45°, 135°60°, 120°

135°, 315°

240°, 300°

30°, 330°

(0, �2.5) (�2.5, 0) or (�2.5, 0), (0, �2.5) depending on direction

3.925 mi (� � 3.14)

x2 � y2 � 6.25, center � (0, 0)

Career – Commercial Fisher

mathmatters3.com/mathworks

http://mathmatters3.com/mathworks

In this book, you have studied a variety of problem solving strategies.Experience in applying these strategies will help you decide which willbe most appropriate for solving a particular problem. Sometimes, onlyone strategy will work. In other cases, any one of several strategies willoffer a solution. There may be times when you will want to use twodifferent approaches to a problem to be sure the solution you found iscorrect. For certain problems, you will need to use more than onestrategy to find the solution.

PRACTICE EXERCISES

Solve. Name the strategy you used to solve each problem. Findlengths to the nearest tenth and angles to the nearest degree.

1. ARCHAEOLOGY The GreatPyramid at El Giza, Egypt,measures 755 ft on a side.The faces stand at a 52° angleto the ground. The top 30 ftof the pyramid has beendestroyed. How tall was thepyramid when it was firstbuilt?

2. The sine of a certain acute angle is equal to the cosine of 24°.Find the acute angle.

3. SURVEYING Two surveyors standing 2.8 mi apart each measure the angle of elevation to the top of a mountain. The surveyor nearer the mountain, standing 3.4 mi from the base of the peak, measures an angle of 26°. Find the angle of elevation measured by the other surveyor.

4. GEOGRAPHY The formula l � 69.81 cos d gives thelength in miles, l, of one degree of longitude on the Earth’s surface, where d is the latitude in degrees.

a. Find the length of one degree of longitude atlatitude 42° N.

b. At what northern hemisphere latitude is the length of one degree of longitude 19.2 mi?

5. SCIENCE This sine wave appeared on a laboratoryoscilloscope screen. The technician then generated a wavecongruent to this one, but 3 units lower on the screen. Findthe equation of the second wave.

14-5 Problem Solving Skills:Choose a Strategy


755 ft

52�

A D C2.8 mi 3.4 mi

B

8

4

�4

�8

90 1800

Problem SolvingStrategies

Guess and check

Look for a pattern

Solve a simplerproblem

Make a table, chartor list

Use a picture,diagram or model

Act it out

Work backwards

Eliminate possibilities

Use an equation orformula

483.2 ft

66°

15°

51.9 mi

74° N

y � 8 sin 2x � 3

GEOGRAPHY The world’s longest aerial ropeway ascends from the city ofMerida, Venezuela (altitude: 5379 ft), to the summit of Pico Espejo (altitude:15,629 ft). The ropeway is 42,240 ft in length.

6. Find the horizontal distance from the lower end of the ropeway to thepoint directly under the summit of Pico Espejo.

7. Find the angle of elevation of the ropeway.

8. PHYSICS A spring bounces up and down in such a way that the height ininches, h, of the weight at the end of the spring is given by h � 6 sin 360t,where t is the number of seconds after the spring reaches the positionshown for the first time.

a. How high will the weight be after 5.1 sec?

b. What is the maximum height reached by the weight?

c. When is the maximum height first reached?

d. What is the minimum height reached by the weight?

e. When is the minimum height first reached?


Round answers to the nearest hundredth if necessary. (Lesson 13-6)

9. If y varies inversely as x, and y � 3 when x � 42, find y when x � 27.






Simplify. (Lesson 11-2)

15. (x � 5)(x � 8) 16. (2x � 7)(3x � 4)

17. (�3x � 2)(�x � 8) 18. (6x � 4)(x � 4)

19. (2x � 9)(2x � 9) 20. 4(x � 3)(x � 4)

21. 3(2x � 1)(x � 6) 22. (4x � 7)(3x � 2)

23. (2x � 9)(3x � 8)

Compute the variance and standard deviation for each set of data. Roundanswers to the nearest hundredth if necessary. (Lesson 9-7)

24. 7, 8, 8, 5, 6, 8, 9, 7 25. 12, 11, 13, 17, 15, 13, 12, 14

26. 21, 23, 20, 25, 25, 29, 27, 26 27. 15, 16, 19, 17, 18, 10, 4, 28

Lesson 14-5 Problem Solving Skills: Choose a Strategy 635

h

t0

40,977.5 ft

14°

3.5 in.

0.25 sec

6 in.

�6 in.

0.75 sec

4.67

5.54

3.75

19.29

3.77

1.42

x2 � 3x � 40

6x2 � 20x � 16

6x2 � 33x � 18

6x2 � 13x � 28

3x2 � 26x � 16

4x2 � 28x � 48

6x2 � 11x � 72

4x2 � 81

12x2 � 29x � 14

1.44; 1.20

8.00; 2.83 42.36; 6.51

3.23; 1.80

Five-step Plan

1 Read2 Plan3 Solve4 Answer5 Check


Chapter 14 ReviewVOCABULARY

Choose the word or phrase from the list that best completes each statement.

1. ___?__ is opposite over adjacent.

2. A(n) ___?__ is formed by a horizontal line and a line slantingupward.

3. ___?__ is adjacent over hypotenuse.

4. A(n) ___?__ is formed by a horizontal line and a line slantingdownward.

5. ___?__ is opposite over hypotenuse.

6. A(n) ___?__ is an acute angle formed by the x-axis and theterminal side of an angle.

7. In a periodic function, the ___?__ is the length of one completecycle of the function.

8. In a periodic function, the ___?__ is half the difference betweenthe maximum and minimum y-values.

9. The sine, cosine, and tangent are three ___?__ ratios.

10. To ___?__ a right triangle means to find the measures of all the parts of the triangle.

LESSON 14-1 Basic Trigonometric Ratios, p. 604

� The ratio of the lengths of two sides of a right triangle depends on the anglesof the triangle.

� For a given angle, the ratios are always the same.

In a right triangle, �A is an acute angle. Then,

sin A � ,

cos A � , and

tan A � .

Use the figure at the right to find each ratio.

11. sin D 12. cos F

13. tan F 14. cos D

15. sin F 16. tan D

17. Visitors to Pittsburgh, Pennsylvania, can ride the Incline from the river valleyup Mt. Washington. The Incline has a 403-ft rise and a 685-ft run. What is theangle made by the track and the horizontal?

length of leg opposite �A��length of leg adjacent to �A

length of leg adjacent to �A��


length of leg opposite �A��


a. amplitude

b. angle of depression

c. angle of elevation

d. cosine

e. initial side

f. period

g. reference angle

h. sine

i. solve

j. tangent

k. terminal side

l. trigonometric

F

D E4

53

Chapter 14 Review 637

LESSON 14-2 Solve Right Triangles, p. 608

� You can find the measures of the angles and sides of a right triangle.

a. Given the measure of one acute angle, find the measure of the other bysubtracting the measure of the known angle from 90°.

b. Given the lengths of two sides, use the Pythagorean Theorem to find thelength of the third side.

c. Given an angle and length of a side, use trigonometric ratios to find otherparts of the triangle.

� Trigonometry problems may involve angles of depression and elevation.

a. An angle of depression is formed by a horizontal line and a line slantingdownward.

b. An angle of elevation is formed by a horizontal line and a line slantingupward.

Find the following in�ABC.

18. BC

19. m�A

20. AC

21. A tower casts a shadow 55 ft long. Measuring from the end of theshadow, Brad determines that the angle of the sun is 43°. How tall is the tower?

22. A kite is flying at the end of a 300-ft string. Assuming the string isstraight and forms an angle of 58° with the ground, how high is the kite?

23. From the top of an observation tower 50 m high, a forest ranger spots a deer at an angle of depression of 28°. How far is the deer from the base of the tower?

LESSON 14-3 Graph the Sine Function, p. 614

� The reference angle is the acute angle formed by the x-axis and the terminal side.

The reference angle measures 210° � 180° � 30°. Draw a triangle with the terminal side as the hypotenuse by making a line perpendicular to the x-axis. The triangle is a 30°-60°-90° right triangle. In this case, the leg lengths are negative because they were drawn by moving in negative directions from the x- and y-axis. The length of the terminal side is always considered positive.

sin 210° � � ��12

�


24. sin 120° 25. tan 225° 26. cos 960°

27. tan 300 28. cos 405° 29. sin 390°

opposite��hypotenuse

y

x

210� initial side

terminalside

referenceangle

43�

55 ft

A

B C

15

35�


LESSON 14-4 Experiment with the Sine Function, p. 618

� Functions with repeating patterns are called periodic functions. The period of a function is the length of one complete cycle of the function.

The period of y � sin x is 360°.

The period of y � sin nx is .

� The amplitude of a periodic function is half the difference between itsmaximum and minimum y-values. It is a measure of the height and depth of a curve.

30. Graph y � 6 sin x. State the amplitude.

31. Graph y � sin �12

�x. State the period.

Tell if the function is periodic. If it is, state the period.

32.

33.

LESSON 14-5 Problem Solving Skills: Choose a Strategy, p. 624

� Experience in applying strategies will help you solve a particular problem. Sometimes one strategy will work. There may be times that you will need to use more than one strategy to find a solution.

34. A steamboat paddlewheel with a radius of 40 in. makes one complete revolution every 4 sec. Half the wheel is submerged. Point P on the rim of the wheel is at the water line. Where in relation to the surface of the water will P be 17.2 sec from now?

35. Astronauts in a lunar lander see a large crater on the moon. The angle of depression to the far side of the crater is 18°. The angle of depression to the near side of the crater is 25°. If the lunar lander is 3 mi above the surface of the moon, what is the distance across the crater?

CHAPTER INVESTIGATION

EXTENSION Make a scale drawing to show how you determined the height ordepth of one of the objects you measured. Label each part of your drawing andinclude all measurements.

O 2 4 6 8 10 12

y

x14 16 18

1

3O 6 9 12 15 18 21 24 27 30 33 36 39

y

x42 45 48 51 54

360°�

n

�360 �180 180 360

1

�1

y

x

P

18�

3 mi

orbit

25�

Chapter 14 AssessmentUse the figure at the right to find each ratio.

1. sin A

2. cos C

3. sin C

4. tan A

5. cos A

6. tan C

Find the following in �CAT.

7. CT to the nearest tenth.

8. m�C

9. CA to the nearest tenth.

10. The General Sherman tree at Sequoia National Park is about 273 ft tall. If you’re standing 64 ft from the base, what is the angle of elevation?

11. GOLF A golfer is standing at the tee, looking up to the green on a hill. If the tee is 36 yd lower than the green and the angle of elevation from the tee to the hole is 12°, find the distance from the tee to the hole.


12. sin 120° 13. cos 240°

14. cos (�45°) 15. tan 300°

16. sin (�780°) 17. tan 405°

Graph. State the period and amplitude.

18. y � �32

� sin x

19. y � cos 3x

20. y � 3 sin x � 2

21. Sound is caused from continuous vibrations. You can think of a sine graphwhen describing sound. When the amplitude of sound vibrations is large, thesound is more intense. Suppose the graph y � 3 sin x describes a soundvibration. To make the sound more intense, the amplitude grows to 8. Writeand graph the equation that describes the intense sound.

12˚36 yd

AC

T

16.5

43�

C

A B8

106

Chapter 14 Assessment 639mathmatters3.com/chapter_assessment

http://mathmatters3.com/chapter_assessment


Standardized Test Practice5. Choose a proportion that can be used to find the

distance across the lake (AB). (Lesson 7-7)

�7A

0B� � �

912

00

�� 87

00� � �

AB120��

�7A

0B� � �

812

00

�� 89

00� � �

12A

0B

��

6. In the following equation, what is the value of c? (Lesson 8-5)

� � � 2� � � � ��9 �6 0 3

7. Which are the solutions of the equation x2 � 7x � 18 � 0? (Lesson 12-3)

2 or �9 �2 or 9

�2 or �9 2 or 9

8. Which value equals cos (�420˚)? (Lesson 14-3)

� ��12�

�12�

�3��

2DC

B�3��

2A

DC

BA

DCBA

bd

ac

�6�4

1�3

68

7�3

DC

BA

80 m

90 m120 m

70 m

73°

73°

59°

59°

A

B

C

D

E

lake

Test-Taking TipQuestion 2The drawings provided for test items are often not drawn toscale. So remember what you can and cannot assume from ageometric figure.You can assume the following from a drawing.• When points appear on a line or line segment, they are

collinear.• Angles that appear to be adjacent or vertical are.• When lines, line segments, or rays appear to intersect,

they do.You cannot assume the following from a drawing.• Line segments or angles that appear to be congruent are.• Lines or line segments that appear to be parallel or

perpendicular are.• A point that appears to be a midpoint of a segment is.

Part 1 Multiple Choice

Record your answers on the answer sheetprovided by your teacher or on a sheet of paper.

1. Which is the simplest form of x7 � x�11?(Lesson 1-8)

�x14� x4 x18

2. In the figure, find the value of x. (Lesson 4-1)

40 50 60 90

3. If you double the length and width of arectangle, how does its perimeter change?(Lesson 5-2)

It does not change.

It is multiplied by 1�12�.

It doubles.

It quadruples.

4. Which is the approximate volume of the cone?(Lesson 5-7)

136.1 mm3

304.3 mm3

408.4 mm3

1021.0 mm3D

C

B

A

13 mm

5 mm

D

C

B

A

DCBA

DCB1

x 18A

x°

60°80°30°

Chapter 14 Standardized Test Practice 641mathmatters3.com/standardized_test

Preparing for Standardized TestsFor test-taking strategies and morepractice, see pages 709–724.

Part 2 Short Response/Grid In

Record your answers on the answer sheetprovided by your teacher or on a sheet of paper.

9. A competition diver receives a score fromeach of 7 judges. The highest and lowestscores are discarded and the remaining scoresare added together. The sum is multiplied bythe dive’s degree of difficulty and thenmultiplied by 0.6. What is the score of a divewith a 1.5-degree of difficulty with thefollowing scores? (Lesson 1-5)

7.5, 7.0, 9.0, 8.5, 6.5, 8.0, 7.0

10. Solve 5(a � 3) � 2a. (Lesson 2-5)

11. In the figure below, find m�BEC.(Lesson 3-2)

12. Find the midpoint of M�Q�. (Lesson 3-3)

13. What is the slope of a line perpendicular to the line represented by the equation 3x � 6y � 12? (Lesson 6-2)

14. If �5x � y � 12 and 5x � 8y � 6, what is thevalue of 9y? (Lesson 6-6)

15. If parallelogram ABCDis reflected over they-axis to become A�B�C�D�,what are the coordinates of C�? (Lesson 8-1)

16. How many ways can 6 swimmers be arrangedon a 4-person relay team? (Lesson 9-5)

y

xC

BD

A

0�1�2 1 2 3�3�4 4

L M N O P Q R S T

DB E

C

(5x � 5)° (2x � 3)°

17. Write (�32�)(�27�) in simplest radical form.(Lesson 10-1)

18. If �ABC is an equilateraltriangle, what is the length ofA�D� in simplestradical form?(Lesson 10-3)

19. If (4x � 2y)(3x � 6y) � ax2 � bxy � cy2, whatis the value of b? (Lesson 11-4)

20. Factor t2 � 14t � 45. (Lesson 11-7)

21. If y varies directly as x2 and y � 100 when x � 5, what is y when x � 7? (Lesson 13-5)

22. For the figure, find cos R. (Lesson 14-1)

Part 3 Extended Response

Record your answers on a sheet of paper.Show your work.

23. A drawbridge is normally 13 ft above thewater. Each section of the drawbridge is 210 ftlong. The angle of elevation of each sectionwhen the bridge is up is 70°. (Lesson 14-2)a. Make a drawing of the situation.b. What trigonometric function would you

use to find the distance from the top of asection of the drawbridge to the waterwhen the bridge is up? Explain why youchose this function.

c. To the nearest foot, what is the distance from the top of a section of the drawbridge to the water when the bridge is up?

24. Graph y � 4 sin 2x. Describe the graphincluding its period and amplitude.(Lesson 14-4)

B 5 C

D

A

45°

S

T

R3

5

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chapter 14: trigonometry

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