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192 PART I: Solutions to Odd-Numbered Exercises and Practice Tests Solutions to Odd.Numbered Exercises 1. adj = ~/5z-3z= ~=4 3. hyp 5 opp 3 sin =0~opp ---- 3 hyp 5 CSC 0-- ~--- cos =0~ adj =-- 4 hyp 5 sec 0 = ~ -’- tan 0 = op___p_p = 3 adj 4 cot 0 = adj = _4 opp 3 hyp 5 adj 4 hyp = ~/82 + 15z = 17 sin 0 = op__p_p _ 8 hyp 17 csc 0 = hy___p_p = 17 opp 8 adj ~5 hyp_ 17 sec 0 = adj - 1-~ cos 0 - hyp 17 tan 0 = op__p_p _ 8 adj 15 adj 15 cot 0 = -opp 8 hyp = ~/i8~ + 12:~ = ~ = 6./~ sin 0 = op___p_p 18 3 3~ hyp = ~~ = "~= 13 cos 0 = adj 12 2 tiyp = ~~ =-~= 13 2.fi-~ tan 0 = op___p.p18 __ 3 adj 12 2 sin 0 = op___p_p -,.. 1 hyp 3 cosO= adj _2 hyp 3 tanO= opp_ 1 adj 2,~- 4 cot0= adj =2 opp 3 csc 0 = hy.___p_p = 3 opp sec 0 = hy___p_p 3 adj - 2.,/~ cot 0 - adj _ 2 opp adj = ,/6z -22 = ,f~ = 4~/~ sin 0 opp =~- 21 hyp 6 3 ¯ cos 0 =~-adj 4,/~ 2~/~ 6 3 hyp 2 1 ,/~ tan 0 = op__p_p adj - 4,f~ - 2~,/~ - 4 hyp 6 csc0- - -3 opp 2 6 3 sec 0 - hyp adj - 4,~/~ - 2~/~ ~ cot 0 adj_ 4~/~ opp 2 - 2,/-~ The function values are the same since the triangles are similar and the corresponding sides are proportional. 193 PART I: Solutions to Odd-Numbered Exercises and Practice Tests opp = ./102 - 82 = 6 sin 0 - opp 6- 3 hyp 10 5 cos 0 - adj 8 -4 hyp 10 5 tan 0 6 3= --opp adj 8 4 0 8 10 5 csc 0 - hyp _ _ opp 6 3 10 5 sec 0 - hyp _ _ adj 8 4 iadj 8cot 4 0- - opp 6 3 opp = ~/2.52 -22 = 1.5 sin 0 =-opp----’=-1.5 3 hyp 2.5 5 hyp 2.5 5 -opp 1.5 3 CSC 0 :---- 2.5 5 -2 4 sec 0 - hyp _ _ hyp 2.5 5 adj 2 4 tan 0 = opp 1.5 3 adj 2 4 cot 0 - - adj 4 opp 1.5 3 2 The function values are the same since the triangles are similar and the corresponding sides are proportional. cos 0 - adj - 11. Given: sin 0 = 5 = op_._p_p 6 hyp 2 2 5 + (adj) = 62 ,cos 0 = adj = .!if tan 0 = op___p_p = adj ./~1 11 cos0= adj=-1 hyp 4 tan 0 = op___p_p = ~ cot 0 -" adj __ -,/i-i opp 5 adj 1 adj 6./]-]" ,~ = 11 ~ 15 cot0= adj _ 1 opp- ~ csc 0 = hy__p_p 4 opp = "~ = 15 csc 0 = hy___p_p 6 opp 5 3 opp 15. Given: tan 0 = 3- - 1 adj 17. Given: cot 0 = _9 = adj 4 opp 32 + 12 hyp sin 0 = sin 0 = op__p_p _ hyp ~ 97 cos0= adj _ 1 hyp- ~cot0= adj_ 1 opp 3 cos0= adj_ 9 hyp ~ sec0=hyp- ~ sec 0 = hy_____p_p_ ~ adj 9 adj csc 0 = hy___p_p = ,/~ opp 3 / / sin 0 = op___p_p _ ~ hyp 4 0~ hyp 6 see 0 = --~ = 6 13. Given: sec 0 = 4 = 4 _ hyp 1 adj (opp)2 + 12 = 42 tan 0 = opp = 4 adj 9 csc 0 = hy___p_p _ opp 4 9.,/~ 97 194 19. sin 60° - 45 PART I: Solutions to Odd-Numbered Exercises and Practice Tests cos 60°=2 2’ (a) tan 60°- sin 60°° _ v/~ cos 60 1 (b) sin 30° = cos 60° 2 (c) cos 30° = sin 60° - 2 ° cos 60 1 ,~ (d) cot 60° sin 60° ,~ - 3 21. cscO= 3, secO- 4 (a) sin 0 = 1_ 1 csc 0 3 ’1 2.,/~ (b) ~ cos 0 - sec 0 - 3 sin 0 1/3 (c) tan 0 - oos 0 - (d) sec(90° - O) = csc 0 = 3 23. cos a = 4 (a) seca- 1 COS O! -4 (b) sin2a + cosZa = 1 sin2 a + sin2 ot = -16 sin a = +~ 4 cos a 1 1/4 ~ (c) cot a- sin a ± .,/~/4- +---~’~/1~ 15 1 (d) sin(90° a) =cosa=4 25. tan Ocot 0 = tan O(ta+O) =1 27. tan a cos a = sin a] cos a = sin a cos a/ 29. (1 + cos 0)(1 - cos O) = 1 - cos2 0 = (sinzO+cos20)-cos20 = Sin2 0 sinO cosO sin:~O+coszO 31. ~ + cos 0 sin 0 sin 0 cos 0 1 sin 0 cos 0 1 1 sin 0 cos 0 = csc 0 sec 0 1 33. (a) cos 60°= 2 195 PART I: Solutions to Odd-Numbered Exercises and Practice Tests 35. (a) cot~-= cgt 45°= 1 (b) cos 45°= ~- 2 39. (a) sin 25° ~ 0.4226 (b) cos 65° = 0.4226 Note: sin 25° = cos(90° - 25°) = cos 65° 37. (a) cos ~ = cos 30° = -~1 (b) sec60°=~=2 ° cos 60 1 41. (a) sec 42° 12’ = sec 42.2° - cos 42.2° -~ 1.3499 1 (b) csc 4807’ sin(48 + ~)o 1.3432 43. Make sure that your calculator is in radian mode. 1 "rr (a) cot 16 - tan(,rr/16) ~ 5.0273 (b) tan ~ = 0.1989 45, Make sure that your calculator is in radian mode. 1 (a) csc 1 sin 1 1.1884 1 (b) tan- ~ 0.5463 2 47. 49. (a) secO=2 =* 0=60°- 3 (a) sin0=2 ~ 0 30° 6 (b) cscO=2 ==, O= 30° 6 51. (a) csc0- 3 ~ 0=60°- 3 (b) sin0 2 ~ 0 45° 4 (b) cot0= 1 ~ 0=45° 4 53. (a) sin 0 = 0.8191 (b) cos 0 = 0.0175 55. (a) tan 0 = 1.1920 :=. 0 = 50° ~ 0.873 radian (b) tan 0 = 0.4663 ~ 0 ~ 25° ~ 0.436 radian 57. tan 30° - y 105 x 59. cot 60°=38 x y = 105 ¯ tan 30° = 105--~- 38 0 = 55° = 0.960 radian 0 ~ 89° = 1.553 radians 196 PART I: Solutions to Odd-Numbered Exercises and Practice-Tests 61. sin 50° = y 15 y- 15 . sin 50°~-, 11.4907 6 h (b) tan 0 = 2- and tan 0 ~ 135 63. (a) 135 ¯ 6 (c) --- h = 270 feet 3 ~ 132-----~ 3 Not drawn to scale 65. tan 0 = op___p_p adj 67. (a) ~3½ w tan 58° = ~ 100 (b) sin 0 - opp hyp w = 100 tan 58° ~ 160.0 feet sin 0 = lO/3- 1 20 6 1 (c) sin 0 = : ~ 0 ~-- 9.59° (d) o ~ 9.59° 69. 71. tan 3° =xM 15 x = 15 tan 3° d=5+2x = 5 + 2(15 tan 3°) ~ 6.57 centimeters opp tan 0 _- --=. adj h tan 80° = -75 h = 75 tan 80° -~ 425.3 meters h cos 0 = adj hyp 75 d COS 80° = -- 1 d = 75 ~ ~ °431.9 meters cos 80 73. x ~ 2.588, y ~ 9.659 y sin 0 = y--- ~ 0.97 10 lO csc o = -- = 1.o4 y cos 0 = -- ~ 0.26 10 x 10 sec 0 =-- = 3.86 tan 0 = y ~ 3.73 cot 0 = x - ~0 0.27 Y x x (x, y) 197 PART I: Solutions to Odd-Numbered Exercises and Practice Tests 10° - csc2 10° = -1 True, because 1 + cotz 0 = csc~ 0 cot20 = csc2 0- 1 77. cot2 75. sec 30° = csc 60° True, because sec(90° - 0) = csc 0. cot2 0 -- CSC~ 0 = --1. 1 20° 0.9397 40° 0.7660 60° 0.5000 80° 0.1736 sin(90°- O) 1 0.9397 0.7660 0.5000 0.1736. 79.I o cos 0 It seems that cos 0 = sin(90° - 0) for all 0. 0 and 90° - 0 are called complementary angles. 83. -3x+8y= 16 Intercepts: (0, 2), (-~q, 0) 81. y= -x-9 Intercepts: (0,-9), (-9, O) Y y 3, ¯ -2, -3-4- 87. - 310° 30’ lies in quadrant I 85. 146° lies in quadrant II Section 4.4 Trigonometric Functions of Any Angle Know the Definitions of Trigonometric Functions of Any Angle~ If 0 is in standard position, (x, y) a point on the terminal side and r = ~/x2 + y2 4= O, then: sinO=-Y r x cos 0 = r cscO= r-, y:i:0 y r sec 0 = -, x g: 0 x cotO=-,y4:oX tano=-Y, x4:0 x y [] You should know the signs of the trigonometric functions in each quadrant. ’n" 3,tr [] You should know the trigonometric function values of the quadrant angles O, ~-, ,rr, and --~-. [] You should be able to find reference angles. [] You should be able to evaluate trigonometric functions of any angle. (Use reference angles.) [] You should know that the period of sine and cosine is 2,tr. You should know which trigonometric functions are odd and even. Even: cos x and sec x Odd: sin x, tan x, cot x, csc x