Solutions for Unit Circle: Sine and Cosine Angles
Solutions to Try Its
1. [latex]\cos \left(t\right)=-\frac{\sqrt{2}}{2},\sin \left(t\right)=\frac{\sqrt{2}}{2}[/latex] 2. [latex]\cos \left(\pi \right)=-1[/latex], [latex]\sin \left(\pi \right)=0[/latex] 3. [latex]\sin \left(t\right)=-\frac{7}{25}[/latex] 4. approximately 0.866025403 5. [latex]\frac{\pi }{3}[/latex] 6. a. [latex]\text{cos}\left(315^\circ \right)=\frac{\sqrt{2}}{2},\text{sin}\left(315^\circ \right)=\frac{-\sqrt{2}}{2}[/latex] b. [latex]\cos \left(-\frac{\pi }{6}\right)=\frac{\sqrt{3}}{2},\sin \left(-\frac{\pi }{6}\right)=-\frac{1}{2}[/latex] 7. [latex]\left(\frac{1}{2},-\frac{\sqrt{3}}{2}\right)[/latex]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, [latex]t[/latex], formed by the terminal side of the angle [latex]t[/latex] and the horizontal axis. 5. The sine values are equal. 7. I 9. IV 11. [latex]\frac{\sqrt{3}}{2}[/latex] 13. [latex]\frac{1}{2}[/latex] 15. [latex]\frac{\sqrt{2}}{2}[/latex] 17. 0 19. −1 21. [latex]\frac{\sqrt{3}}{2}[/latex] 23. [latex]60^\circ [/latex] 25. [latex]80^\circ [/latex] 27. [latex]45^\circ [/latex] 29. [latex]\frac{\pi }{3}[/latex] 31. [latex]\frac{\pi }{3}[/latex] 33. [latex]\frac{\pi }{8}[/latex] 35. [latex]60^\circ [/latex], Quadrant IV, [latex]\text{sin}\left(300^\circ \right)=-\frac{\sqrt{3}}{2},\cos \left(300^\circ \right)=\frac{1}{2}[/latex] 37. [latex]45^\circ [/latex], Quadrant II, [latex]\text{sin}\left(135^\circ \right)=\frac{\sqrt{2}}{2}[/latex], [latex]\cos \left(135^\circ \right)=-\frac{\sqrt{2}}{2}[/latex] 39. [latex]60^\circ [/latex], Quadrant II, [latex]\text{sin}\left(120^\circ \right)=\frac{\sqrt{3}}{2}[/latex], [latex]\cos \left(120^\circ \right)=-\frac{1}{2}[/latex] 41. [latex]30^\circ [/latex], Quadrant II, [latex]\text{sin}\left(150^\circ \right)=\frac{1}{2}[/latex], [latex]\cos \left(150^\circ \right)=-\frac{\sqrt{3}}{2}[/latex] 43. [latex]\frac{\pi }{6}[/latex], Quadrant III, [latex]\text{sin}\left(\frac{7\pi }{6}\right)=-\frac{1}{2}[/latex], [latex]\text{cos}\left(\frac{7\pi }{6}\right)=-\frac{\sqrt{3}}{2}[/latex] 45. [latex]\frac{\pi }{4}[/latex], Quadrant II, [latex]\text{sin}\left(\frac{3\pi }{4}\right)=\frac{\sqrt{2}}{2}[/latex], [latex]\cos \left(\frac{4\pi }{3}\right)=-\frac{\sqrt[]{2}}{2}[/latex] 47. [latex]\frac{\pi }{3}[/latex], Quadrant II, [latex]\text{sin}\left(\frac{2\pi }{3}\right)=\frac{\sqrt{3}}{2}[/latex], [latex]\cos \left(\frac{2\pi }{3}\right)=-\frac{1}{2}[/latex] 49. [latex]\frac{\pi }{4}[/latex], Quadrant IV, [latex]\text{sin}\left(\frac{7\pi }{4}\right)=-\frac{\sqrt{2}}{2}[/latex], [latex]\text{cos}\left(\frac{7\pi }{4}\right)=\frac{\sqrt{2}}{2}[/latex] 51. [latex]\frac{\sqrt{77}}{9}[/latex] 53. [latex]-\frac{\sqrt{15}}{4}[/latex] 55. [latex]\left(-10,10\sqrt{3}\right)[/latex] 57. [latex]\left(-2.778,15.757\right)[/latex] 59. [latex]\left[-1,1\right][/latex] 61. [latex]\sin t=\frac{1}{2},\cos t=-\frac{\sqrt{3}}{2}[/latex] 63. [latex]\sin t=-\frac{\sqrt{2}}{2},\cos t=-\frac{\sqrt{2}}{2}[/latex] 65. [latex]\sin t=\frac{\sqrt{3}}{2},\cos t=-\frac{1}{2}[/latex] 67. [latex]\sin t=-\frac{\sqrt{2}}{2},\cos t=\frac{\sqrt{2}}{2}[/latex] 69. [latex]\sin t=0,\cos t=-1[/latex] 71. [latex]\sin t=-0.596,\cos t=0.803[/latex] 73. [latex]\sin t=\frac{1}{2},\cos t=\frac{\sqrt{3}}{2}[/latex] 75. [latex]\sin t=-\frac{1}{2},\cos t=\frac{\sqrt{3}}{2}[/latex] 77. [latex]\sin t=0.761,\cos t=-0.649[/latex] 79. [latex]\sin t=1,\cos t=0[/latex] 81. −0.1736 83. 0.9511 85. −0.7071 87. −0.1392 89. −0.7660 91. [latex]\frac{\sqrt{2}}{4}[/latex] 93. [latex]-\frac{\sqrt{6}}{4}[/latex] 95. [latex]\frac{\sqrt{2}}{4}[/latex] 97. [latex]\frac{\sqrt{2}}{4}[/latex] 99. 0 101. [latex]\left(0,-1\right)[/latex] 103. 37.5 seconds, 97.5 seconds, 157.5 seconds, 217.5 seconds, 277.5 seconds, 337.5 secondsLicenses & Attributions
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- Precalculus. Provided by: OpenStax Authored by: OpenStax College. Located at: https://cnx.org/contents/[email protected]:1/Preface. License: CC BY: Attribution.
