Solutions for the Other Trigonometric Functions
Solutions to Try Its
1. [latex]\sin t=-\frac{\sqrt{2}}{2},\cos t=\frac{\sqrt{2}}{2},\tan t=-1,\sec t=\sqrt{2},\csc t=-\sqrt{2},\cot t=-1[/latex] 2. [latex]\begin{array}{l}\sin \frac{\pi }{3}=\frac{\sqrt{3}}{2}\\ \cos \frac{\pi }{3}=\frac{1}{2}\\ \tan \frac{\pi }{3}=\sqrt{3}\\ \sec \frac{\pi }{3}=2\\ \csc \frac{\pi }{3}=\frac{2\sqrt{3}}{3}\\ \cot \frac{\pi }{3}=\frac{\sqrt{3}}{3}\end{array}[/latex] 3. [latex]\sin \left(\frac{-7\pi }{4}\right)=\frac{\sqrt{2}}{2},\cos \left(\frac{-7\pi }{4}\right)=\frac{\sqrt{2}}{2},\tan \left(\frac{-7\pi }{4}\right)=1[/latex], [latex-display]\sec \left(\frac{-7\pi }{4}\right)=\sqrt{2},\csc \left(\frac{-7\pi }{4}\right)=\sqrt{2},\cot \left(\frac{-7\pi }{4}\right)=1[/latex-display] 4. [latex]-\sqrt{3}[/latex] 5. [latex]-2[/latex] 6. [latex]\sin t[/latex] 7. [latex]\cos t=-\frac{8}{17},\sin t=\frac{15}{17},\tan t=-\frac{15}{8}[/latex] [latex-display]\csc t=\frac{17}{15},\cot t=-\frac{8}{15}[/latex-display] 8. [latex]\begin{array}{l}\sin t=-1,\cos t=0,\tan t=\text{Undefined}\\ \sec t=\text{\hspace{0.17em}Undefined,}\csc t=-1,\cot t=0\end{array}[/latex] 9. [latex]\sec t=\sqrt{2},\csc t=\sqrt{2},\tan t=1,\cot t=1[/latex] 10. [latex]\approx -2.414[/latex]Solutions to Odd-Numbered Exercises
1. Yes, when the reference angle is [latex]\frac{\pi }{4}[/latex] and the terminal side of the angle is in quadrants I and III. Thus, at [latex]x=\frac{\pi }{4},\frac{5\pi }{4}[/latex], the sine and cosine values are equal. 3. Substitute the sine of the angle in for [latex]y[/latex] in the Pythagorean Theorem [latex]{x}^{2}+{y}^{2}=1[/latex]. Solve for [latex]x[/latex] and take the negative solution. 5. The outputs of tangent and cotangent will repeat every [latex]\pi [/latex] units. 7. [latex]\frac{2\sqrt{3}}{3}[/latex] 9. [latex]\sqrt{3}[/latex] 11. [latex]\sqrt{2}[/latex] 13. 1 15. 2 17. [latex]\frac{\sqrt{3}}{3}[/latex] 19. [latex]-\frac{2\sqrt{3}}{3}[/latex] 21. [latex]\sqrt{3}[/latex] 23. [latex]-\sqrt{2}[/latex] 25. −1 27. −2 29. [latex]-\frac{\sqrt{3}}{3}[/latex] 31. 2 33. [latex]\frac{\sqrt{3}}{3}[/latex] 35. −2 37. −1 39. If [latex]\sin t=-\frac{2\sqrt{2}}{3},\sec t=-3,\csc t=-\frac{3\sqrt{2}}{4},\tan t=2\sqrt{2},\cot t=\frac{\sqrt{2}}{4}[/latex] 41. [latex]\sec t=2,\csc t=\frac{2\sqrt{3}}{3},\tan t=\sqrt{3},\cot t=\frac{\sqrt{3}}{3}[/latex] 43. [latex]-\frac{\sqrt{2}}{2}[/latex] 45. 3.1 47. 1.4 49. [latex]\sin t=\frac{\sqrt{2}}{2},\cos t=\frac{\sqrt{2}}{2},\tan t=1,\cot t=1,\sec t=\sqrt{2},\csc t=\sqrt{2}[/latex] 51. [latex]\sin t=-\frac{\sqrt{3}}{2},\cos t=-\frac{1}{2},\tan t=\sqrt{3},\cot t=\frac{\sqrt{3}}{3},\sec t=-2,\csc t=-\frac{2\sqrt{3}}{3}[/latex] 53. –0.228 55. –2.414 57. 1.414 59. 1.540 61. 1.556 63. [latex]\sin \left(t\right)\approx 0.79[/latex] 65. [latex]\csc t\approx 1.16[/latex] 67. even 69. even 71. [latex]\frac{\sin t}{\cos t}=\tan t[/latex] 73. 13.77 hours, period: [latex]1000\pi [/latex] 75. 7.73 inchesLicenses & 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.
