142 Chapter 2 Trigonometric Functions

. -I~xercis~ S~t 2.:i .- -- .,.-: -~ -'~ -, - :-. - - I ,::,o, - ''''T"':~:. -~:-"r: ~~ .;.~ '. . _ .,_ l.j'. _ - ... _. _ -~.. __ _ _ , _

In Exercises 1 to 12, find the values of the six trigonometric In Exercises 16 to 18, let 0 be an acute angle of a right trian­functIons of (J for the right triangle with the given sides.

gle for which tan 0 = :. Find 2.1.

16. sin (J 17. cot e ) 18. sec e

7

In Exercises 19 to 21, let fJ be an acute angle of a right

triangle for which sec {J = ~~. Rnd

12

35 19. cos {3 20. cot f3 21.cscl3

3'n 4'~3 In Exercises 22 to 24, let () be an acute angle of a right tri­? 4

angle for which cos 8 = ~. Rnd9

22. sin 8 23. sec e 24. tan e 5. J> 6.

In Exercises 25 to 38, find the exact value of each expression.

25. sin 45° + cos 450

27. sin 300 cos 60° - tan 45°

7'{3~ 30. sec 30" cos 30° ­

2 31 . 7T 7T . sm 3 + COS 6

33. sin!!.. + tan!!"6 9 10. ~,9.~ 4 63

O'~

tan 60° cot 60°

'IT" 7T32. csc- - sec­

6 3

34. sin!!.. cos 1T .- tan ~ 344

1T 1T 1T 7T 7T 1T 36. COS "'4 tan 6 + 2 tan 3" 37. 2csc- - sec-cos­

12' 43611. ~ -.[2 0

1 7T 7T 7T38. 3 tlln - + sec - sin ­5~ 4 6 3n

6

:'V In Exercises 39 to 50, use a calculator to find the value of the trigonometric function to four decimal places. In Exercises 13 to 15, let 9 be an acute angle of a right

40. sec 88° 41. cos ('}"20' triangle for which sin (J = ~ .Find

43. cos 34.7" 44. tan 81.30

13. tan e 14. sec e 15. cos e

..-----------------------------------~---_ .. __ .... ~

143

7F 46. sin-;: 47. tan~

J 7

49. csc 1.2 50. sin 0.45

51. VERTICAL HEIGHT FROM SLANT HEIGHT A 12-100t ladder is resting against a wall and makes an angle of 52° with the growld. Find the height to which the ladder will reach on the waU.

52. DISTANCE ACROSS A MARSH Find the distance AB across the marsh shown in the accompanying figure.

53. WIDTH OF A RAMP A skateboarder wishes to build a jump ramp that is inclined at a 19° angle and that has a maxi­mum height of 32.0 inches. Find the horizontal width x of the ramp.

64. TIME OF CLOSEST ApPROACH At 3:00 P.M., a boat is 12.5 miles due west of a radar station and traveling at 11 mph in a direction that is 57.3" south of an east-west line. At what time will the boat be closest to t.he radar station?

2.2 Right Triangle Trigonometry

55. PLACEMENT OF A LIGHT For best illumination of a piece of art, iI lighting spedalist for an art gall.cry fecom­mends that a ceiling-mounted light be 6 feet from the pieCt: of art and that the angle of depression of the light be 38°. How far from a wall should uhe light be placed so that the recommendations of the specialist ace met? Notice that the art extends outward 4 inches from the walL

56. HEIGHT OF THE EJFFEL TOWER The angle of elevation from a point 116 meters from the base of the Eiffel Tower to the top of the tower is 68.9". Find the approximate height of the tower.

57. DISTANCE OF A DESCENT An airplane traveling at 240 mph is descending at an angle of depression of 6°. How many miles will the plane descend in 4 minutes? .

») 58. TIME OF A DeSCENT A submarine traveling at 9.0 mph is desceJ1djng at an angle of depression of SO. How many minutes, to the nearest tenth. does it take the submarine to reach a depth of 80 feet?

59. HEIGHT OF A BUILDING A surveyor determines that the

angle of elevation from a transit to the top of a building is 27.8°. The transit is positioned 5.5 feet above ground level and 131 feel from the building. Find the height of the building to the nearest tenth of a foot.

TrallSil

\ \ 27.8'

5.5/t

))lli-----l._-L-..-lI'I...Eil.«l

144 Chapter 2 Trigonometric Functions

60. WIDTH OF A LAKE The angle of depression to one side of a lake, measured from a balloDn 2500 feel above the lake as shown in the accompanying figure, is 43'. The angle of depression to the opposite side of the lake is 27°. Pind the width of the lake.

63. AREA OF AN ISOSCELES TRIANGLE Consider the following isosceles triangle. TIle length of each of the two equal sides of he triangle is (1, and each of the base angles has a measure of O. Verify that the' area of the triangle is A = (12 sin 0 cos O.

61. ASTRONOMY The moon Europa rotates in a nearly circular orbit around Jupiter. The orbital radius of Europa is approximately 670,900 kilometers. During a revolution of Europa around Jupiter, an astronomer found that the maximum value of the angle () formed by Europa, Earth, and Jupiter was 0.056°. Find the dis­tance d between Earth and Jupiter at the lime the as­tronomer found the maximum value of O. Round to the nearest million kilometers.

Not drawn to scale.

62. ASTRONOMY Venus rotates in a nearly circular orbit around the sun. The largest angle formed by Venus, Earth, and the sun is 46.5". The di!stance from 'Earth to the sun is approximately 149,000,000 kilometers. See the follOWing figure. What is the orbital radius r of Venus? Round 10 Ihe nearest million kilometers.

a

o

64. AREA OF A HEXAGON Find the area of the hexagon. (Hint: The area consists of six isosceles mangles. Use the formula from Exercise 63 J

, ,

to compute the area of one of the 4in./ \4m.

mangles and multiply by 6.) I \

':if{). 60"/'

4fn.

65. HEIGHT OF A PYRAMID The angle of eleva lion to the top of the Egyptian pyramid of Cheops is 36.4°, measured from a point 3.'i0 feet from the base of the pyramid. The angle of devation from the base of a face at the pyramid is 51.9°. Find the height of the Cheops pyramid.

2.2 Right Triangle Trigonometry 145

66. HEIGHT OF A BUILDING 'FwD buildings are 240 feet apart. TIle aI1lgle of elevation from the top of the shorter b~lilding

to the top of the other building ,is 22°.lf the shorter build­ing is 80 feet high, how high is the tailer building?

67. HEIGHT OF THE WASHINGTON MONUMENT From a point A on a line from the base of the Washington Monument, the angle of elevation to the top of the monument is 42.0°. F.rom a point 100 feet away from A and on the same line, the angle to the top is 37.8°. Find the height, to the nearest foot, of the Washington Monument.

connected by a skybridge at Ihe forty-firsll1oor. Note the information given in the accompanying figure.

a. Determine the height of the skybridge.

b. Determine the length of the skybridge.

AB ;: 412 feet LCAB =53.6'

AB i~ at ground level LCAD =15S

68. HEIGHT OF A TOWER The angle of elevation from a point A to the top of a tower is 32.1°. From point B, which is on the same line but 55.5 feet closer to the tower, the angle of eJevation is 36.7". Find the height of the tower.

A

69. THE PETAONAS TOWERS The Petronas Towers in Kuala Lumpur, MaJay~ia, are the world's tallest twin towers. Each tower is 1483 feet in height. The towers are

70. AN EIFFEL TOWER REPLICA Use the inlormation in the accompanying figure to estimate the height of the Eiffel Tower replica that stands in front of the Paris Las Vegas Hotel in Las Vegas, Nevada.

46.30