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Обучение Руководства > Precalculus II

Solutions for Unit Circle: Sine and Cosine Angles

Solutions to Try Its

1. cos⁡(t)=−22,sin⁡(t)=22\cos \left(t\right)=-\frac{\sqrt{2}}{2},\sin \left(t\right)=\frac{\sqrt{2}}{2} 2. cos⁡(π)=−1\cos \left(\pi \right)=-1, sin⁡(π)=0\sin \left(\pi \right)=0 3. sin⁡(t)=−725\sin \left(t\right)=-\frac{7}{25} 4. approximately 0.866025403 5. π3\frac{\pi }{3} 6. a. cos(315∘)=22,sin(315∘)=−22\text{cos}\left(315^\circ \right)=\frac{\sqrt{2}}{2},\text{sin}\left(315^\circ \right)=\frac{-\sqrt{2}}{2} b. cos⁡(−π6)=32,sin⁡(−π6)=−12\cos \left(-\frac{\pi }{6}\right)=\frac{\sqrt{3}}{2},\sin \left(-\frac{\pi }{6}\right)=-\frac{1}{2} 7. (12,−32)\left(\frac{1}{2},-\frac{\sqrt{3}}{2}\right)

Solutions to Odd-Numbered Exercises

1. The unit circle is a circle of radius 1 centered at the origin. 3. Coterminal angles are angles that share the same terminal side. A reference angle is the size of the smallest acute angle, tt, formed by the terminal side of the angle tt and the horizontal axis. 5. The sine values are equal. 7. I 9. IV 11. 32\frac{\sqrt{3}}{2} 13. 12\frac{1}{2} 15. 22\frac{\sqrt{2}}{2} 17. 0 19. −1 21. 32\frac{\sqrt{3}}{2} 23. 60∘60^\circ 25. 80∘80^\circ 27. 45∘45^\circ 29. π3\frac{\pi }{3} 31. π3\frac{\pi }{3} 33. π8\frac{\pi }{8} 35. 60∘60^\circ , Quadrant IV, sin(300∘)=−32,cos⁡(300∘)=12\text{sin}\left(300^\circ \right)=-\frac{\sqrt{3}}{2},\cos \left(300^\circ \right)=\frac{1}{2} 37. 45∘45^\circ , Quadrant II, sin(135∘)=22\text{sin}\left(135^\circ \right)=\frac{\sqrt{2}}{2}, cos⁡(135∘)=−22\cos \left(135^\circ \right)=-\frac{\sqrt{2}}{2} 39. 60∘60^\circ , Quadrant II, sin(120∘)=32\text{sin}\left(120^\circ \right)=\frac{\sqrt{3}}{2}, cos⁡(120∘)=−12\cos \left(120^\circ \right)=-\frac{1}{2} 41. 30∘30^\circ , Quadrant II, sin(150∘)=12\text{sin}\left(150^\circ \right)=\frac{1}{2}, cos⁡(150∘)=−32\cos \left(150^\circ \right)=-\frac{\sqrt{3}}{2} 43. π6\frac{\pi }{6}, Quadrant III, sin(7π6)=−12\text{sin}\left(\frac{7\pi }{6}\right)=-\frac{1}{2}, cos(7π6)=−32\text{cos}\left(\frac{7\pi }{6}\right)=-\frac{\sqrt{3}}{2} 45. π4\frac{\pi }{4}, Quadrant II, sin(3π4)=22\text{sin}\left(\frac{3\pi }{4}\right)=\frac{\sqrt{2}}{2}, cos⁡(4π3)=−22\cos \left(\frac{4\pi }{3}\right)=-\frac{\sqrt[]{2}}{2} 47. π3\frac{\pi }{3}, Quadrant II, sin(2π3)=32\text{sin}\left(\frac{2\pi }{3}\right)=\frac{\sqrt{3}}{2}, cos⁡(2π3)=−12\cos \left(\frac{2\pi }{3}\right)=-\frac{1}{2} 49. π4\frac{\pi }{4}, Quadrant IV, sin(7π4)=−22\text{sin}\left(\frac{7\pi }{4}\right)=-\frac{\sqrt{2}}{2}, cos(7π4)=22\text{cos}\left(\frac{7\pi }{4}\right)=\frac{\sqrt{2}}{2} 51. 779\frac{\sqrt{77}}{9} 53. −154-\frac{\sqrt{15}}{4} 55. (−10,103)\left(-10,10\sqrt{3}\right) 57. (−2.778,15.757)\left(-2.778,15.757\right) 59. [−1,1]\left[-1,1\right] 61. sin⁡t=12,cos⁡t=−32\sin t=\frac{1}{2},\cos t=-\frac{\sqrt{3}}{2} 63. sin⁡t=−22,cos⁡t=−22\sin t=-\frac{\sqrt{2}}{2},\cos t=-\frac{\sqrt{2}}{2} 65. sin⁡t=32,cos⁡t=−12\sin t=\frac{\sqrt{3}}{2},\cos t=-\frac{1}{2} 67. sin⁡t=−22,cos⁡t=22\sin t=-\frac{\sqrt{2}}{2},\cos t=\frac{\sqrt{2}}{2} 69. sin⁡t=0,cos⁡t=−1\sin t=0,\cos t=-1 71. sin⁡t=−0.596,cos⁡t=0.803\sin t=-0.596,\cos t=0.803 73. sin⁡t=12,cos⁡t=32\sin t=\frac{1}{2},\cos t=\frac{\sqrt{3}}{2} 75. sin⁡t=−12,cos⁡t=32\sin t=-\frac{1}{2},\cos t=\frac{\sqrt{3}}{2} 77. sin⁡t=0.761,cos⁡t=−0.649\sin t=0.761,\cos t=-0.649 79. sin⁡t=1,cos⁡t=0\sin t=1,\cos t=0 81. −0.1736 83. 0.9511 85. −0.7071 87. −0.1392 89. −0.7660 91. 24\frac{\sqrt{2}}{4} 93. −64-\frac{\sqrt{6}}{4} 95. 24\frac{\sqrt{2}}{4} 97. 24\frac{\sqrt{2}}{4} 99. 0 101. (0,−1)\left(0,-1\right) 103. 37.5 seconds, 97.5 seconds, 157.5 seconds, 217.5 seconds, 277.5 seconds, 337.5 seconds

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