Precalculus Student Solution Chapter 5

120 Chapter 5 Try It 5.1. 5.2. 3π 2 5.3. −135° 121 5.4. 7π 10 5.5. α = 150 ° 5.6. β = 60 ° 5.7. 7π 6 5.8. 215π = 37.525 units 18 5.9. 1.88 5.10. −3π 2 rad/s 5.11. 1655 kilometers per hour 5.12. cos(t) = − 2 , sin(t) = 2 2 2 5.13. cos(π) = − 1, sin(π) = 0 5.14. sin(t) = − 7 25 5.15. approximately 0.866025403 5.16. π 3 5.17. a. cos(315 ° ) = 2 , sin(315 ° ) = – 2 2 2 b. 5.18. ⎞ ⎛ cos( − π ) = 3 , sin⎝− π ⎠ = − 1 2 2 6 6 ⎛1 3⎞ ⎝2 , − 2 ⎠ 5.19. sin t = − 2 , cos t = 2 , tan t = − 1, sec t = 2, csc t = − 2, cot t = − 1 2 2 5.20. sin π = 3 3 2 π 1 cos = 3 2 π tan = 3 3 sec π = 2 3 π csc = 2 3 3 3 cot π = 3 3 3 122 5.21. ⎞ ⎞ ⎞ ⎛ ⎛ ⎛ sin⎝−7π ⎠ = 2 , cos⎝−7π ⎠ = 2 , tan⎝−7π ⎠ = 1, 2 2 4 4 4 ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ sec⎝−7π ⎠ = 2, csc⎝−7π ⎠ = 2, cot⎝−7π ⎠ = 1 4 4 4 5.22. − 3 5.23. −2 5.24. sin t 5.25. cos t = − 8 , sin t = 15 , tan t = − 15 17 17 8 csc t = 17 , cot t = − 8 15 15 5.26. sin t = − 1, cos t = 0, tan t = Undefine sec t = Undefined csc t = − 1, cot t = 0 5.27. sec t = 2, csc t = 2, tan t = 1, cot t = 1 5.28. ≈ − 2.414 5.29. 7 25 5.30. sin t = 33 , cos t = 56 , tan t = 33 , 65 65 56 65 65 sec t = , csc t = , cot t = 56 33 33 56 5.31. ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ sin⎝π ⎠ = 2 , cos⎝π ⎠ = 2 , tan⎝π ⎠ = 1, 2 2 4 4 4 ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ sec⎝π ⎠ = 2, csc⎝π ⎠ = 2, cot⎝π ⎠ = 1 4 4 4 5.32. 2 5.33. adjacent = 10; opposite = 10 3 ; missing angle is π 6 5.34. About 52 ft Section Exercises 1. 123 3. Whether the angle is positive or negative determines the direction. A positive angle is drawn in the counterclockwise direction, and a negative angle is drawn in the clockwise direction. 5. Linear speed is a measurement found by calculating distance of an arc compared to time. Angular speed is a measurement found by calculating the angle of an arc compared to time. 7. 9. 11. 124 13. 15. 17. 240° 125 19. 4π 3 21. 2π 3 23. 7π ≈ 11.00 in 2 2 25. 81π ≈ 12.72 cm 2 20 27. 20° 126 29. 60° 31. −75° 33. π 2 radians 35. −3π radians 37. π radians 39. 5π 6 radians 41. 5.02π ≈ 5.26 3 miles 43. 25π ≈ 8.73 9 centimeters 45. 21π ≈ 6.60 10 meters 47. 104.7198 cm2 49. 0.7697 in2 51. 250° 53. 320° 55. 4π 3 57. 8π 9 59. 1320 rad 210.085 RPM 61. 7 in./s, 4.77 RPM, 28.65 deg/s 63. 1, 809, 557.37 mm/min = 30.16 m/s 65. 5.76 miles 67. 120 ° 69. 794 miles per hour 71. 2,234 miles per hour 73. 11.5 inches 75. The unit circle is a circle of radius 1 centered at the origin. 77. Coterminal angles are angles that share the same terminal side. A reference angle is the size of the smallest acute angle, t, formed by the terminal side of the angle t and the horizontal axis. 79. The sine values are equal. 81. I 83. IV 85. 3 2 87. 1 2 89. 2 2 91. 0 127 93. −1 95. 3 2 97. 60° 99. 80° 101. 45° 103. π 3 105. π 3 107. π 8 109. 60°, Quadrant IV, sin(300°) = − 3 , cos(300°) = 1 2 2 111. 45°, Quadrant II, sin(135°) = 2 , 2 cos(135°) = − 2 2 113. 60°, Quadrant II, sin(120°) = 3 , 2 cos(120°) = − 1 2 115. 30°, Quadrant II, sin(150°) = 1 , 2 cos(150°) = − 3 2 117. π, 6 Quadrant III, sin( 7π ) = − 1 , 2 6 cos( 7π ) = − 3 2 6 119. π, 4 Quadrant II, sin( 3π ) = 2 , 4 2 ⎛4π ⎞ cos⎝ ⎠ = − 2 3 2 121. π, 3 Quadrant II, 128 sin( 2π ) = 3 , 3 2 ⎛2π ⎞ cos⎝ ⎠ = − 1 3 2 123. π, 4 Quadrant IV, sin( 7π ) = − 2 , 2 4 7π 2 cos( ) = 4 2 125. 77 9 127. − 15 4 129. −10, 10 3⎞⎠ ⎛ ⎝ 131. ( – 2.778, 15.757) 133. [ – 1, 1] 135. sint = 1 , cost = − 3 2 2 137. sin t = − 2 , cos t = − 2 2 2 139. sin t = 3 , cos t = − 1 2 2 141. sin t = − 2 , cos t = 2 2 2 143. sin t = 0, cos t = − 1 145. sin t = − 0.596, cos t = 0.803 147. sin t = 1 , cos t = 3 2 2 149. sin t = − 1 , cos t = 3 2 2 151. sin t = 0.761, cos t = − 0.649 153. sin t = 1, cos t = 0 155. −0.1736 157. 0.9511 159. −0.7071 161. −0.1392 163. −0.7660 129 165. 2 4 167. − 6 4 169. 2 4 171. 2 4 173. 0 175. (0, – 1) 177. 37.5 seconds, 97.5 seconds, 157.5 seconds, 217.5 seconds, 277.5 seconds, 337.5 seconds 179. Yes, when the reference angle is π 4 and the terminal side of the angle is in quadrants I and III. Thus, at x = π , 5π , 4 4 the sine and cosine values are equal. 181. Substitute the sine of the angle in for y in the Pythagorean Theorem x 2 + y 2 = 1. Solve for x and take the negative solution. 183. The outputs of tangent and cotangent will repeat every π units. 185. 2 3 3 187. 3 189. 2 191. 1 193. 2 195. 3 3 197. −2 3 3 199. 3 201. − 2 203. −1 205. −2 207. − 3 3 209. 2 211. 3 3 213. −2 215. −1 130 217. If sin t = − 2 2 , sec t = − 3, csc t = − 3 2 , tan t = 2 2, cot t = 2 3 4 4 219. sec t = 2, csc t = 2 3 , tan t = 3, cot t = 3 3 3 221. − 2 2 223. 3.1 225. 1.4 227. sin t = 2 , cos t = 2 , tan t = 1, cot t = 1, sec t = 2, csc t = 2 2 2 229. sin t = − 3 , cos t = − 1 , tan t = 3, cot t = 3 , sec t = − 2, csc t = − 2 3 2 3 3 2 231. –0.228 233. –2.414 235. 1.414 237. 1.540 239. 1.556 241. sin(t) ≈ 0.79 243. csct ≈ 1.16 245. even 247. even 249. sin t cos t = tan t 251. 13.77 hours, period: 1000π 253. 7.73 inches 255. 257. The tangent of an angle is the ratio of the opposite side to the adjacent side. 259. For example, the sine of an angle is equal to the cosine of its complement; the cosine of an angle is equal to the sine of its complement. 261. π 6 263. π 4 265. b = 20 3 , c = 40 3 3 3 267. a = 10, 000, c = 10, 000.5 131 269. b = 5 3 , c = 10 3 3 3 271. 5 29 29 273. 5 2 275. 29 2 277. 5 41 41 279. 5 4 281. 41 4 283. c = 14, 285. a = 15, b=7 3 b = 15 287. b = 9.9970, 289. a = 2.0838, c = 12.2041 b = 11.8177 291. a = 55.9808, c = 57.9555 293. a = 46.6790, b = 17.9184 295. a = 16.4662, c = 16.8341 297. 188.3159 299. 200.6737 301. 498.3471 ft 303. 1060.09 ft 305. 27.372 ft 307. 22.6506 ft 309. 368.7633 ft Review Exercises 311. 45° 313. − 7π 6 315. 10.385 meters 317. 60° 319. 2π 11 321. 132 323. 325. 1036.73 miles per hour 327. 3 2 329. –1 331. π 4 333. − 2 2 335. [ – 1, 1] 337. 1 339. 2 341. 2 343. 0.6 345. 2 2 or − 2 2 347. sine, cosecant, tangent, cotangent 133 349. 3 3 351. 0 353. b = 8, c = 10 355. 11 157 157 357. a = 4, b=4 359. 14.0954 ft Practice Test 361. 150° 363. 6.283 centimeters 365. 15° 367. 369. 3.351 feet per second, 2π 75 radians per second 371. − 3 2 373. [ – 1, 1] 375. 3 377. 3 3 379. 3 2 381. π 3 383. a = 9, b = 9 3 2 2