"

Section 8.1 Answers

  1. [latex]\csc(\theta) = \sqrt{5}[/latex]
  2. [latex]\cos(\theta) = -\frac{1}{4}[/latex]
  3. [latex]\cot(t) = \frac{1}{3}[/latex]
  4. [latex]\sin(\theta) = -\frac{12}{13}[/latex]
  5. [latex]\sec(\theta) = -\sqrt{5}[/latex]
  6. [latex]\csc(t) = \sqrt{5}[/latex]
  7. [latex]\tan(\theta) = -2\sqrt{2}[/latex]
  8. [latex]\cos(\theta) = -\frac{\sqrt{5}}{3}[/latex]
  9. [latex]\cos(t) \approx 0.9075[/latex]
  10. [latex]\tan(\theta) \approx - 0.6074[/latex]
  11. [latex]\csc(t) \approx -4.079[/latex]
  12. [latex]\sin(\theta) = \frac{3}{5}, \cos(\theta) = -\frac{4}{5}, \tan(\theta) = -\frac{3}{4}, \csc(\theta) = \frac{5}{3}, \sec(\theta) = -\frac{5}{4}, \cot(\theta) = -\frac{4}{3}[/latex]
  13. [latex]\sin(\theta) = -\frac{12}{13}, \cos(\theta) = -\frac{5}{13}, \tan(\theta) = \frac{12}{5}, \csc(\theta) = -\frac{13}{12}, \sec(\theta) = -\frac{13}{5}, \cot(\theta) = \frac{5}{12}[/latex]
  14. [latex]\sin(\theta) = \frac{24}{25}, \cos(\theta) = \frac{7}{25}, \tan(\theta) = \frac{24}{7}, \csc(\theta) = \frac{25}{24}, \sec(\theta) = \frac{25}{7}, \cot(\theta) = \frac{7}{24}[/latex]
  15. [latex]\sin(\theta) = \frac{-4\sqrt{3}}{7}, \cos(\theta) = \frac{1}{7}, \tan(\theta) = -4\sqrt{3}, \csc(\theta) = -\frac{7\sqrt{3}}{12}, \sec(\theta) = 7, \cot(\theta) = -\frac{\sqrt{3}}{12}[/latex]
  16. [latex]\sin(\theta) = -\frac{\sqrt{91}}{10}, \cos(\theta) = -\frac{3}{10}, \tan(\theta) = \frac{\sqrt{91}}{3}, \csc(\theta) = -\frac{10\sqrt{91}}{91}, \sec(\theta) = -\frac{10}{3}, \cot(\theta) = \frac{3\sqrt{91}}{91}[/latex]
  17. [latex]\sin(\theta) = \frac{\sqrt{530}}{530}, \cos(\theta) = -\frac{23\sqrt{530}}{530}, \tan(\theta) = -\frac{1}{23}, \csc(\theta) = \sqrt{530}, \sec(\theta) = -\frac{\sqrt{530}}{23}, \cot(\theta) = -23[/latex]
  18. [latex]\sin(\theta) = -\frac{2\sqrt{5}}{5}, \cos(\theta) = \frac{\sqrt{5}}{5}, \tan(\theta) = -2, \csc(\theta) = -\frac{\sqrt{5}}{2}, \sec(\theta) = \sqrt{5}, \cot(\theta) = -\frac{1}{2}[/latex]
  19. [latex]\sin(\theta) = \frac{\sqrt{15}}{4}, \cos(\theta) = -\frac{1}{4}, \tan(\theta) = -\sqrt{15}, \csc(\theta) = \frac{4\sqrt{15}}{15}, \sec(\theta) = -4, \cot(\theta) = -\frac{\sqrt{15}}{15}[/latex]
  20. [latex]\sin(\theta) = -\frac{\sqrt{6}}{6}, \cos(\theta) = -\frac{\sqrt{30}}{6}, \tan(\theta) = \frac{\sqrt{5}}{5}, \csc(\theta) = -\sqrt{6}, \sec(\theta) = -\frac{\sqrt{30}}{5}, \cot(\theta) = \sqrt{5}[/latex]
  21. [latex]\sin(\theta) = \frac{2\sqrt{2}}{3}, \cos(\theta) = \frac{1}{3}, \tan(\theta) = 2\sqrt{2}, \csc(\theta) = \frac{3\sqrt{2}}{4}, \sec(\theta) = 3, \cot(\theta) = \frac{\sqrt{2}}{4}[/latex]
  22. [latex]\sin(t) = \frac{\sqrt{5}}{5}, \cos(t) = \frac{2\sqrt{5}}{5}, \tan(t) = \frac{1}{2}, \csc(t) = \sqrt{5}, \sec(t) = \frac{\sqrt{5}}{2}, \cot(t) = 2[/latex]
  23. [latex]\sin(t) = \frac{1}{5}, \cos(t) = -\frac{2\sqrt{6}}{5}, \tan(t) = -\frac{\sqrt{6}}{12}, \csc(t) = 5, \sec(t) = -\frac{5\sqrt{6}}{12}, \cot(t) = -2\sqrt{6}[/latex]
  24. [latex]\sin(t) = -\frac{\sqrt{110}}{11}, \cos(t) = -\frac{\sqrt{11}}{11}, \tan(t) = \sqrt{10}, \csc(t) = -\frac{\sqrt{110}}{10}, \sec(t) = -\sqrt{11}, \cot(t) = \frac{\sqrt{10}}{10}[/latex]
  25. [latex]\sin(t) = -\frac{\sqrt{95}}{10}, \cos(t) = \frac{\sqrt{5}}{10}, \tan(t) = -\sqrt{19}, \csc(t) = -\frac{2\sqrt{95}}{19}, \sec(t) = 2\sqrt{5}, \cot(t) = -\frac{\sqrt{19}}{19}[/latex]
  26. No, Skippy is not correct. In order to be an identity, an equation must hold for all applicable angles. For example, [latex]\cos(\theta) + \sin(\theta) = 1[/latex] does not hold when [latex]\theta = \pi[/latex].

Section 8.2 Answers

  1. Answer May Vary
  2. Answer May Vary
  3. Answer May Vary
  4. Answer May Vary
  5. Answer May Vary
  6. Answer May Vary
  7. [latex]\cos(75^{\circ}) = \frac{\sqrt{6} - \sqrt{2}}{4}[/latex]
  8. [latex]\sec(165^{\circ}) = -\frac{4}{\sqrt{2}+\sqrt{6}} = \sqrt{2} - \sqrt{6}[/latex]
  9. [latex]\sin(105^{\circ}) = \frac{\sqrt{6}+\sqrt{2}}{4}[/latex]
  10. [latex]\csc(195^{\circ}) = \frac{4}{\sqrt{2}-\sqrt{6}} = -(\sqrt{2}+\sqrt{6})[/latex]
  11. [latex]\cot(255^{\circ}) = \frac{\sqrt{3}-1}{\sqrt{3}+1} = 2-\sqrt{3}[/latex]
  12. [latex]\tan(375^{\circ}) = \frac{3-\sqrt{3}}{3+\sqrt{3}} = 2-\sqrt{3}[/latex]
  13. [latex]\cos\left(\frac{13\pi}{12}\right) = -\frac{\sqrt{6}+\sqrt{2}}{4}[/latex]
  14. [latex]\sin\left(\frac{11\pi}{12}\right) = \frac{\sqrt{6} - \sqrt{2}}{4}[/latex]
  15. [latex]\tan\left(\frac{13\pi}{12}\right) = \frac{3-\sqrt{3}}{3+\sqrt{3}} = 2-\sqrt{3}[/latex]
  16. [latex]\cos \left( \frac{7\pi}{12} \right) = \frac{\sqrt{2} - \sqrt{6}}{4}[/latex]
  17. [latex]\tan \left( \frac{17\pi}{12} \right) = 2 + \sqrt{3}[/latex]
  18. [latex]\sin \left( \frac{\pi}{12} \right) = \frac{\sqrt{6} - \sqrt{2}}{4}[/latex]
  19. [latex]\cot \left( \frac{11\pi}{12} \right) = -(2 + \sqrt{3})[/latex]
  20. [latex]\csc \left( \frac{5\pi}{12} \right) = \sqrt{6} - \sqrt{2}[/latex]
  21. [latex]\sec \left( -\frac{\pi}{12} \right) = \sqrt{6} - \sqrt{2}[/latex]
  22. .
    1. [latex]\cos(\alpha + \beta) = -\frac{\sqrt{2}}{10}[/latex]
    2. [latex]\sin(\alpha + \beta) = \frac{7\sqrt{2}}{10}[/latex]
    3. [latex]\tan(\alpha + \beta) = -7[/latex]
    4. [latex]\cos(\alpha - \beta)= -\frac{\sqrt{2}}{2}[/latex]
    5. [latex]\sin(\alpha - \beta) = \frac{\sqrt{2}}{2}[/latex]
    6. [latex]\tan(\alpha - \beta) = -1[/latex]
  23. .
    1. [latex]\cos(\alpha + \beta) = - \frac{4+7\sqrt{2}}{30}[/latex]
    2. [latex]\sin(\alpha + \beta) = \frac{28-\sqrt{2}}{30}[/latex]
    3. [latex]\tan(\alpha + \beta) = \frac{-28+\sqrt{2}}{4+7\sqrt{2}} = \frac{63-100\sqrt{2}}{41}[/latex]
    4. [latex]\cos(\alpha - \beta) = \frac{-4+7\sqrt{2}}{30}[/latex]
    5. [latex]\sin(\alpha - \beta) = - \frac{28+\sqrt{2}}{30}[/latex]
    6. [latex]\tan(\alpha - \beta)= \frac{28+\sqrt{2}}{4-7\sqrt{2}} = -\frac{63+100\sqrt{2}}{41}[/latex]
  24. .
    1. [latex]\sin(\alpha + \beta) = \frac{16}{65}[/latex]
    2. [latex]\cos(\alpha - \beta) = \frac{33}{65}[/latex]
    3. [latex]\tan(\alpha - \beta) = \frac{56}{33}[/latex]
  25. .
    1. [latex]\csc(\alpha - \beta) = -\frac{5}{4}[/latex]
    2. [latex]\sec(\alpha + \beta) = \frac{125}{117}[/latex]
    3. [latex]\cot(\alpha + \beta) = \frac{117}{44}[/latex]
  26. [latex]f(t) = \sqrt{2}\sin(t) + \sqrt{2}\cos(t) + 1 = 2\sin\left(t + \frac{\pi}{4}\right) + 1 = 2\cos\left(t + \frac{7\pi}{4}\right) + 1[/latex]
  27. [latex]f(t) = 3\sqrt{3}\sin(3t) - 3\cos(3t) = 6\sin\left(3t + \frac{11\pi}{6}\right) = 6\cos\left(3t + \frac{4\pi}{3}\right)[/latex]
  28. [latex]f(t) = -\sin(t) + \cos(t) - 2 = \sqrt{2}\sin\left(t + \frac{3\pi}{4}\right) - 2 = \sqrt{2}\cos\left(t + \frac{\pi}{4}\right) - 2[/latex]
  29. [latex]f(t) = -\frac{1}{2}\sin(2t) - \frac{\sqrt{3}}{2}\cos(2t) = \sin\left(2t + \frac{4\pi}{3}\right) = \cos\left(2t + \frac{5\pi}{6}\right)[/latex]
  30. [latex]f(t) = 2\sqrt{3} \cos(t) - 2\sin(t) = 4\sin\left(t+\frac{2\pi}{3} \right) = 4\cos\left(t + \frac{\pi}{6}\right)[/latex]
  31. [latex]f(t) = \frac{3}{2} \cos(2t) - \frac{3\sqrt{3}}{2} \sin(2t) + 6 =3\sin\left(2t + \frac{5\pi}{6}\right) + 6 = 3\cos\left(2t + \frac{\pi}{3}\right) + 6[/latex]
  32. [latex]f(t) = -\frac{1}{2} \cos(5t) -\frac{\sqrt{3}}{2} \sin(5t) = \sin\left(5t + \frac{7\pi}{6}\right) = \cos\left(5t + \frac{2\pi}{3}\right)[/latex]
  33. [latex]f(t) = -6\sqrt{3} \cos(3t) - 6\sin(3t) - 3 = 12\sin\left(3t + \frac{4\pi}{3}\right) - 3 = 12\cos\left(3t + \frac{5\pi}{6}\right) - 3[/latex]
  34. [latex]f(t) = \frac{5\sqrt{2}}{2} \sin(t) -\frac{5\sqrt{2}}{2} \cos(t) = 5\sin\left(t + \frac{7\pi}{4}\right)= 5\cos\left(t + \frac{5\pi}{4}\right)[/latex]
  35. [latex]f(t) =3\sin\left(\frac{t}{6}\right) -3\sqrt{3} \cos\left(\frac{t}{6}\right) = 6\sin\left( \frac{t}{6}+\frac{5\pi}{3}\right)= 6\cos\left( \frac{t}{6}+\frac{7\pi}{6}\right)[/latex]
  36. [latex]\cos(75^{\circ}) = \frac{\sqrt{2-\sqrt{3}}}{2}[/latex]
  37. [latex]\sin(105^{\circ}) = \frac{\sqrt{2+\sqrt{3}}}{2}[/latex]
  38. [latex]\cos(67.5^{\circ}) = \frac{\sqrt{2-\sqrt{2}}}{2}[/latex]
  39. [latex]\sin(157.5^{\circ}) = \frac{\sqrt{2-\sqrt{2}}}{2}[/latex]
  40. [latex]\tan(112.5^{\circ}) = - \sqrt{\frac{2+\sqrt{2}}{2-\sqrt{2}}} = -1 - \sqrt{2}[/latex]
  41. [latex]\cos\left( \frac{7\pi}{12} \right) = -\frac{\sqrt{2-\sqrt{3}}}{2}[/latex]
  42. [latex]\sin\left( \frac{\pi}{12} \right) = \frac{\sqrt{2-\sqrt{3}}}{2}[/latex]
  43. [latex]\cos \left( \frac{\pi}{8} \right) = \frac{\sqrt{2 + \sqrt{2}}}{2}[/latex]
  44. [latex]\sin \left( \frac{5\pi}{8} \right) = \frac{\sqrt{2 + \sqrt{2}}}{2}[/latex]
  45. [latex]\tan \left( \frac{7\pi}{8} \right) = -\sqrt{ \frac{2 - \sqrt{2}}{2 + \sqrt{2}} } =1-\sqrt{2}[/latex]
  46. .
    • [latex]\sin(2\theta) = -\frac{336}{625}[/latex]
    • [latex]\sin\left(\frac{\theta}{2}\right) = \frac{\sqrt{2}}{10}[/latex]
    • [latex]\cos(2\theta) = \frac{527}{625}[/latex]
    • [latex]\cos\left(\frac{\theta}{2}\right) = -\frac{7\sqrt{2}}{10}[/latex]
    • [latex]\tan(2\theta) = -\frac{336}{527}[/latex]
    • [latex]\tan\left(\frac{\theta}{2}\right) = -\frac{1}{7}[/latex]
  47. .
    • [latex]\sin(2\theta) = \frac{2520}{2809}[/latex]
    • [latex]\sin\left(\frac{\theta}{2}\right) = \frac{5\sqrt{106}}{106}[/latex]
    • [latex]\cos(2\theta) = -\frac{1241}{2809}[/latex]
    • [latex]\cos\left(\frac{\theta}{2}\right) = \frac{9\sqrt{106}}{106}[/latex]
    • [latex]\tan(2\theta) = -\frac{2520}{1241}[/latex]
    • [latex]\tan\left(\frac{\theta}{2}\right) = \frac{5}{9}[/latex]
  48. .
    • [latex]\sin(2\theta) = \frac{120}{169}[/latex]
    • [latex]\sin\left(\frac{\theta}{2}\right) = \frac{3\sqrt{13}}{13}[/latex]
    • [latex]\cos(2\theta) = -\frac{119}{169}[/latex]
    • [latex]\cos\left(\frac{\theta}{2}\right) = -\frac{2\sqrt{13}}{13}[/latex]
    • [latex]\tan(2\theta) = -\frac{120}{119}[/latex]
    • [latex]\tan\left(\frac{\theta}{2}\right) = -\frac{3}{2}[/latex]
  49. .
    • [latex]\sin(2\theta) = -\frac{\sqrt{15}}{8}[/latex]
    • [latex]\sin\left(\frac{\theta}{2}\right) =\frac{\sqrt{8+2\sqrt{15}}}{4}[/latex]
    • [latex]\cos(2\theta) = \frac{7}{8}[/latex]
    • [latex]\cos\left(\frac{\theta}{2}\right) = \frac{\sqrt{8-2\sqrt{15}}}{4}[/latex]
    • [latex]\tan(2\theta) = -\frac{\sqrt{15}}{7}[/latex]
    • [latex]\tan\left(\frac{\theta}{2}\right) = \sqrt{\frac{8+2\sqrt{15}}{8-2\sqrt{15}}}[/latex]
  50. .
    • [latex]\sin(2\theta) = \frac{24}{25}[/latex]
    • [latex]\sin\left(\frac{\theta}{2}\right) = \frac{\sqrt{5}}{5}[/latex]
    • [latex]\cos(2\theta) = -\frac{7}{25}[/latex]
    • [latex]\cos\left(\frac{\theta}{2}\right) = \frac{2\sqrt{5}}{5}[/latex]
    • [latex]\tan(2\theta)=-\frac{24}{7}[/latex]
    • [latex]\tan\left(\frac{\theta}{2}\right) = \frac{1}{2}[/latex]
  51. .
    • [latex]\sin(2\theta) = \frac{24}{25}[/latex]
    • [latex]\sin\left(\frac{\theta}{2}\right) = \frac{2\sqrt{5}}{5}[/latex]
    • [latex]\cos(2\theta) = -\frac{7}{25}[/latex]
    • [latex]\cos\left(\frac{\theta}{2}\right) = -\frac{\sqrt{5}}{5}[/latex]
    • [latex]\tan(2\theta)=-\frac{24}{7}[/latex]
    • [latex]\tan\left(\frac{\theta}{2}\right) = -2[/latex]
  52. .
    • [latex]\sin(2\theta) = -\frac{120}{169}[/latex]
    • [latex]\sin\left(\frac{\theta}{2}\right) = \frac{\sqrt{26}}{26}[/latex]
    • [latex]\cos(2\theta) = \frac{119}{169}[/latex]
    • [latex]\cos\left(\frac{\theta}{2}\right) = -\frac{5\sqrt{26}}{26}[/latex]
    • [latex]\tan(2\theta)=-\frac{120}{119}[/latex]
    • [latex]\tan\left(\frac{\theta}{2}\right) = -\frac{1}{5}[/latex]
  53. .
    • [latex]\sin(2\theta) = -\frac{120}{169}[/latex]
    • [latex]\sin\left(\frac{\theta}{2}\right) = \frac{5\sqrt{26}}{26}[/latex]
    • [latex]\cos(2\theta) = \frac{119}{169}[/latex]
    • [latex]\cos\left(\frac{\theta}{2}\right) = \frac{\sqrt{26}}{26}[/latex]
    • [latex]\tan(2\theta)=-\frac{120}{119}[/latex]
    • [latex]\tan\left(\frac{\theta}{2}\right) = 5[/latex]
  54. .
    • [latex]\sin(2\theta) = -\frac{4}{5}[/latex]
    • [latex]\sin\left(\frac{\theta}{2}\right) = \frac{\sqrt{50-10\sqrt{5}}}{10}[/latex]
    • [latex]\cos(2\theta) = -\frac{3}{5}[/latex]
    • [latex]\cos\left(\frac{\theta}{2}\right)= -\frac{\sqrt{50+10\sqrt{5}}}{10}[/latex]
    • [latex]\tan(2\theta)=\frac{4}{3}[/latex]
    • [latex]\tan\left(\frac{\theta}{2}\right) = -\sqrt{\frac{5-\sqrt{5}}{5+\sqrt{5}}}[/latex]
  55. .
    • [latex]\sin(2\theta) = -\frac{4}{5}[/latex]
    • [latex]\sin\left(\frac{\theta}{2}\right) = \frac{\sqrt{50+10\sqrt{5}}}{10}[/latex]
    • [latex]\cos(2\theta) = -\frac{3}{5}[/latex]
    • [latex]\cos\left(\frac{\theta}{2}\right)= \frac{\sqrt{50-10\sqrt{5}}}{10}[/latex]
    • [latex]\tan(2\theta)=\frac{4}{3}[/latex]
    • [latex]\tan\left(\frac{\theta}{2}\right) = \sqrt{\frac{5+\sqrt{5}}{5-\sqrt{5}}}[/latex]
  56. [latex]\tan(t) = \frac{x}{\sqrt{1 - x^2}}[/latex]
  57. [latex]\sec(\theta) = \sqrt{1+x^2}[/latex]
  58. [latex]\tan(\theta) = -\sqrt{x^2-1}[/latex]
  59. [latex]\cos(2t) = 1 - \dfrac{x^{2}}{2}[/latex]
  60. [latex]\sin(2\theta) = \dfrac{14x}{x^{2} + 49}[/latex]
  61. [latex]\ln|\sec(t) + \tan(t)| = \ln |x + \sqrt{x^{2} + 16}| - \ln(4)[/latex]
  62. [latex]\dfrac{\cos(2\theta) + \cos(8\theta)}{2}[/latex]
  63. [latex]\dfrac{\cos(5t) - \cos(9t)}{2}[/latex]
  64. [latex]\dfrac{\sin(8x) + \sin(10x)}{2}[/latex]
  65. [latex]\dfrac{\cos(4\theta) + \cos(8\theta)}{2}[/latex]
  66. [latex]\dfrac{\cos(t) - \cos(5t)}{2}[/latex]
  67. [latex]\dfrac{\sin(2x) + \sin(4x)}{2}[/latex]
  68. [latex]2\cos(4\theta)\cos(\theta)[/latex]
  69. [latex]-2\cos \left( \frac{9}{2}t \right) \sin \left( \frac{5}{2}t \right)[/latex]
  70. [latex]2\sin \left( \frac{11}{2}x \right) \sin \left( \frac{1}{2}x \right)[/latex]
  71. [latex]2\cos(4\theta)\sin(5\theta)[/latex]
  72. [latex]\sqrt{2}\cos \left(t - \frac{\pi}{4} \right)[/latex]
  73. [latex]-\sqrt{2}\sin \left(x - \frac{\pi}{4} \right)[/latex]
  74. [latex]f(t) = [2\cos(t)] \cos(4t)[/latex], [latex]A(t) = 2\cos(t)[/latex], wave-envelope: [latex]y = \pm 2\cos(t)[/latex]
  75. [latex]f(t) = \left[6\sin \left( \frac{1}{2} t \right) \right] \sin \left( \frac{11}{2} t \right)[/latex], [latex]A(t) = 6\sin \left( \frac{1}{2} t \right)[/latex], wave-envelope: [latex]y = \pm 6\sin \left( \frac{1}{2} t \right)[/latex]
  76. [latex]f(t) = [\cos(4t)] \sin(5t)[/latex], [latex]A(t) = \cos(4t)[/latex], wave-envelope: [latex]y = \pm \cos(4t)[/latex]
  77. [latex]f(t) = \left[-\frac{4}{3}\sin \left( \frac{5}{2} t \right) \right]\cos \left( \frac{9}{2} t \right)[/latex], [latex]A(t) = -\frac{4}{3}\sin \left( \frac{5}{2} t \right)[/latex], wave-envelope: [latex]y = \pm \frac{4}{3} \sin \left( \frac{5}{2} t \right)[/latex].

Section 8.3 Answers

  1. [latex]\theta = \arcsin\left(\dfrac{7}{11}\right) + 2\pi k[/latex] or [latex]\theta = \pi - \arcsin\left(\dfrac{7}{11}\right) + 2\pi k[/latex], in [latex][0, 2\pi)[/latex], [latex]\theta \approx 0.6898, \, 2.4518[/latex]
  2. [latex]\theta = \arccos\left(-\dfrac{2}{9}\right) + 2\pi k[/latex] or [latex]\theta = - \arccos\left(-\dfrac{2}{9}\right) + 2\pi k[/latex], in [latex][0, 2\pi)[/latex], [latex]\theta \approx 1.7949, \, 4.4883[/latex]
  3. [latex]\theta = \pi + \arcsin(0.569) + 2\pi k[/latex] or [latex]\theta = 2\pi - \arcsin(0.569) + 2\pi k[/latex], in [latex][0, 2\pi)[/latex], [latex]\theta \approx 3.7469, \, 5.6779[/latex]
  4. [latex]\theta= \arccos(0.117) + 2\pi k[/latex] or [latex]\theta = 2\pi - \arccos(0.117) + 2\pi k[/latex], in [latex][0, 2\pi)[/latex], [latex]\theta \approx 1.4535, \, 4.8297[/latex]
  5. [latex]\theta = \arcsin(0.008) + 2\pi k[/latex] or [latex]\theta = \pi - \arcsin(0.008) + 2\pi k[/latex], in [latex][0, 2\pi)[/latex], [latex]\theta \approx 0.0080, \, 3.1336[/latex]
  6. [latex]\theta = \arccos\left(\dfrac{359}{360}\right) + 2\pi k[/latex] or [latex]\theta = 2\pi - \arccos\left(\dfrac{359}{360}\right) + 2\pi k[/latex], in [latex][0, 2\pi)[/latex], [latex]\theta \approx 0.0746, \, 6.2086[/latex]
  7. [latex]t = \arctan(117) + \pi k[/latex], in [latex][0, 2\pi)[/latex], [latex]t \approx 1.56225, \, 4.70384[/latex]
  8. [latex]t = \arctan\left(-\dfrac{1}{12}\right) + \pi k[/latex], in [latex][0, 2\pi)[/latex], [latex]t \approx 3.0585, \, 6.2000[/latex]
  9. [latex]t = \arccos\left(\dfrac{2}{3}\right) + 2\pi k[/latex] or [latex]t = 2\pi - \arccos\left(\dfrac{2}{3}\right) + 2\pi k[/latex], in [latex][0, 2\pi)[/latex], [latex]t \approx 0.8411, \, 5.4422[/latex]
  10. [latex]t = \pi + \arcsin\left(\dfrac{17}{90}\right) + 2\pi k[/latex] or [latex]t = 2\pi - \arcsin\left(\dfrac{17}{90}\right) + 2\pi k[/latex], in [latex][0, 2\pi)[/latex], [latex]t \approx 3.3316, \, 6.0932[/latex]
  11. [latex]t = \arctan\left(-\sqrt{10}\right) + \pi k[/latex], in [latex][0, 2\pi)[/latex], [latex]t \approx 1.8771, \, 5.0187[/latex]
  12. [latex]t = \arcsin\left(\dfrac{3}{8}\right) + 2\pi k[/latex] or [latex]t = \pi - \arcsin\left(\dfrac{3}{8}\right) + 2\pi k[/latex], in [latex][0, 2\pi)[/latex], [latex]t \approx 0.3844, \, 2.7572[/latex]
  13. [latex]x = \arccos\left(-\dfrac{7}{16}\right) + 2\pi k[/latex] or [latex]x = - \arccos\left(-\dfrac{7}{16}\right) + 2\pi k[/latex], in [latex][0, 2\pi)[/latex], [latex]x \approx 2.0236, \, 4.2596[/latex]
  14. [latex]x = \arctan(0.03) + \pi k[/latex], in [latex][0, 2\pi)[/latex], [latex]x \approx 0.0300, \, 3.1716[/latex]
  15. [latex]x = \arcsin(0.3502) + 2\pi k[/latex] or [latex]x = \pi - \arcsin(0.3502) + 2\pi k[/latex], in [latex][0, 2\pi)[/latex], [latex]x \approx 0.3578, \,2.784[/latex]
  16. [latex]x = \pi + \arcsin(0.721) + 2\pi k[/latex] or [latex]x = 2\pi - \arcsin(0.721) + 2\pi k[/latex], in [latex][0, 2\pi)[/latex], [latex]x \approx 3.9468, \, 5.4780[/latex]
  17. [latex]x = \arccos(0.9824) + 2\pi k[/latex] or [latex]x = 2\pi - \arccos(0.9824) + 2\pi k[/latex], in [latex][0, 2\pi)[/latex], [latex]x \approx 0.1879, \, 6.0953[/latex]
  18. [latex]x = \arccos(-0.5637) + 2\pi k[/latex] or [latex]x = - \arccos(-0.5637) + 2\pi k[/latex], in [latex][0, 2\pi)[/latex], [latex]x \approx 2.1697, \, 4.1135[/latex]
  19. [latex]x = \arctan(117) + \pi k[/latex], in [latex][0, 2\pi)[/latex], [latex]x \approx 1.5622, \, 4.7038[/latex]
  20. [latex]x = \arctan(-0.6109) + \pi k[/latex], in [latex][0, 2\pi)[/latex], [latex]x \approx 2.5932, \, 5.7348[/latex]
  21. [latex]\theta = \dfrac{\pi k}{5}; \; \theta = 0, \dfrac{\pi}{5}, \dfrac{2\pi}{5}, \dfrac{3\pi}{5}, \dfrac{4\pi}{5}, \pi, \dfrac{6\pi}{5}, \dfrac{7\pi}{5}, \dfrac{8\pi}{5}, \dfrac{9\pi}{5}[/latex]
  22. [latex]t = \dfrac{\pi}{9} + \dfrac{2\pi k}{3}[/latex] or [latex]t = \dfrac{5\pi}{9} + \dfrac{2\pi k}{3}; \; t = \dfrac{\pi}{9}, \dfrac{5\pi}{9}, \dfrac{7\pi}{9}, \dfrac{11\pi}{9}, \dfrac{13\pi}{9}, \dfrac{17\pi}{9}[/latex]
  23. [latex]x = \dfrac{2\pi}{3} + \pi k[/latex] or [latex]x = \dfrac{5\pi}{6} + \pi k; \; x = \dfrac{2\pi}{3}, \dfrac{5\pi}{6}, \dfrac{5\pi}{3}, \dfrac{11\pi}{6}[/latex]
  24. [latex]\theta = \dfrac{\pi}{24} + \dfrac{\pi k}{6}; \; \theta = \dfrac{\pi}{24}, \dfrac{5\pi}{24}, \dfrac{3\pi}{8}, \dfrac{13\pi}{24}, \dfrac{17\pi}{24}, \dfrac{7\pi}{8}, \dfrac{25\pi}{24}, \dfrac{29\pi}{24}, \dfrac{11\pi}{8}, \dfrac{37\pi}{24}, \dfrac{41\pi}{24}, \dfrac{15\pi}{8}[/latex]
  25. [latex]t = \dfrac{3\pi}{8} + \dfrac{\pi k}{2}; \; t = \dfrac{3\pi}{8}, \dfrac{7\pi}{8}, \dfrac{11\pi}{8}, \dfrac{15\pi}{8}[/latex]
  26. [latex]x = \dfrac{\pi}{12} + \dfrac{2\pi k}{3}[/latex] or [latex]x = \dfrac{7\pi}{12} + \dfrac{2\pi k}{3}; \; x = \dfrac{\pi}{12}, \dfrac{7\pi}{12}, \dfrac{3\pi}{4}, \dfrac{5\pi}{4}, \dfrac{17\pi}{12}, \dfrac{23\pi}{12}[/latex]
  27. [latex]\theta = \dfrac{\pi}{3} + \dfrac{\pi k}{2}; \; \theta = \dfrac{\pi}{3}, \dfrac{5\pi}{6}, \dfrac{4\pi}{3}, \dfrac{11\pi}{6}[/latex]
  28. No solution
  29. [latex]x = \dfrac{3\pi}{4} + 6\pi k[/latex] or [latex]x = \dfrac{9\pi}{4} + 6\pi k; \; x = \dfrac{3\pi}{4}[/latex]
  30. [latex]\theta = -\dfrac{\pi}{3} + \pi k; \; \theta = \dfrac{2\pi}{3}, \dfrac{5\pi}{3}[/latex]
  31. [latex]t = \dfrac{3\pi}{4} + \pi k[/latex] or [latex]t = \dfrac{13\pi}{12} + \pi k; \; t = \dfrac{\pi}{12}, \dfrac{3\pi}{4}, \dfrac{13\pi}{12}, \dfrac{7\pi}{4}[/latex]
  32. [latex]x = -\dfrac{19\pi}{12} + 2\pi k[/latex] or [latex]x = \dfrac{\pi}{12} + 2\pi k; \; x = \dfrac{\pi}{12}, \dfrac{5\pi}{12}[/latex]
  33. No solution
  34. [latex]t = \dfrac{5\pi}{8} + \dfrac{\pi k}{2}; \; t = \dfrac{\pi}{8}, \dfrac{5\pi}{8}, \dfrac{9\pi}{8}, \dfrac{13\pi}{8}[/latex]
  35. [latex]x = \dfrac{\pi}{3} + \pi k[/latex] or [latex]x = \dfrac{2\pi}{3} + \pi k; \; x = \dfrac{\pi}{3}, \dfrac{2\pi}{3}, \dfrac{4\pi}{3}, \dfrac{5\pi}{3}[/latex]
  36. [latex]\theta = \dfrac{\pi}{6} + \pi k[/latex] or [latex]\theta = \dfrac{5\pi}{6} + \pi k; \; \theta = \dfrac{\pi}{6}, \dfrac{5\pi}{6}, \dfrac{7\pi}{6}, \dfrac{11\pi}{6}[/latex]
  37. [latex]t = \dfrac{\pi}{4} + \dfrac{\pi k}{2}; \; t = \dfrac{\pi}{4}, \dfrac{3\pi}{4}, \dfrac{5\pi}{4}, \dfrac{7\pi}{4}[/latex]
  38. [latex]x = \dfrac{\pi}{3} + \pi k[/latex] or [latex]x = \dfrac{2\pi}{3} + \pi k; \; x = \dfrac{\pi}{3}, \dfrac{2\pi}{3}, \dfrac{4\pi}{3}, \dfrac{5\pi}{3}[/latex]
  39. [latex]\theta = \dfrac{\pi}{4}, \dfrac{5\pi}{4}[/latex]
  40. [latex]t = 0, \dfrac{\pi}{3}, \pi, \dfrac{5\pi}{3}[/latex]
  41. [latex]x = \dfrac{\pi}{6}, \dfrac{\pi}{2}, \dfrac{5\pi}{6}, \dfrac{3\pi}{2}[/latex]
  42. [latex]\theta = \dfrac{\pi}{6}, \dfrac{5\pi}{6}, \dfrac{3\pi}{2}[/latex]
  43. [latex]t = 0, \dfrac{2\pi}{3}, \dfrac{4\pi}{3}[/latex]
  44. [latex]x=\dfrac{\pi}{3}, \dfrac{5\pi}{3}[/latex]
  45. [latex]\theta = \dfrac{2\pi}{3}, \dfrac{4\pi}{3}, \arccos\left(\dfrac{1}{3}\right), 2\pi -\arccos\left(\dfrac{1}{3}\right)[/latex]
  46. [latex]t=\dfrac{\pi}{6}, \dfrac{5\pi}{6}[/latex]
  47. [latex]x = \dfrac{7\pi}{6}, \dfrac{11\pi}{6}, \arcsin\left(\dfrac{1}{3}\right), \pi - \arcsin\left(\dfrac{1}{3}\right)[/latex]
  48. [latex]\theta=\dfrac{3\pi}{4}, \dfrac{7\pi}{4}, \arctan\left(\dfrac{1}{2}\right), \pi +\arctan\left(\dfrac{1}{2}\right)[/latex]
  49. [latex]t=0, \dfrac{2\pi}{3}, \dfrac{4\pi}{3}[/latex]
  50. [latex]x=\dfrac{\pi}{6}, \dfrac{5\pi}{6}, \dfrac{\pi}{2}[/latex]
  51. [latex]\theta=\arctan(2), \pi + \arctan(2)[/latex]
  52. [latex]t = \dfrac{\pi}{6}, \dfrac{7\pi}{6}, \dfrac{5\pi}{6}, \dfrac{11\pi}{6}[/latex]
  53. [latex]x = 0, \pi, \dfrac{\pi}{4}, \dfrac{3\pi}{4}, \dfrac{5\pi}{4}, \dfrac{7\pi}{4}[/latex]
  54. [latex]\theta = \dfrac{\pi}{6}, \dfrac{\pi}{4}, \dfrac{3\pi}{4}, \dfrac{5\pi}{6}, \dfrac{7\pi}{6}, \dfrac{5\pi}{4}, \dfrac{7\pi}{4}, \dfrac{11\pi}{6}[/latex]
  55. [latex]t = \dfrac{\pi}{2}, \dfrac{3\pi}{2}[/latex]
  56. [latex]x = 0, \dfrac{\pi}{3}, \dfrac{2\pi}{3}, \pi, \dfrac{4\pi}{3}, \dfrac{5\pi}{3}[/latex]
  57. [latex]\theta = \dfrac{\pi}{3}, \dfrac{5\pi}{3}[/latex]
  58. [latex]t = \dfrac{\pi}{2}, \dfrac{3\pi}{2}[/latex]
  59. [latex]x = \dfrac{\pi}{6}, \dfrac{\pi}{2}, \dfrac{5\pi}{6}, \dfrac{3\pi}{2}[/latex]
  60. [latex]\theta = \dfrac{\pi}{6}, \dfrac{5\pi}{6}, \dfrac{7\pi}{6}, \dfrac{3\pi}{2}, \dfrac{11\pi}{6}[/latex]
  61. [latex]t = \dfrac{\pi}{8}, \dfrac{5\pi}{8}, \dfrac{9\pi}{8}, \dfrac{13\pi}{8}[/latex]
  62. No solution
  63. [latex]\theta = 0, \dfrac{\pi}{7}, \dfrac{2\pi}{7}, \dfrac{3\pi}{7}, \dfrac{4\pi}{7}, \dfrac{5\pi}{7}, \dfrac{6\pi}{7}, \pi, \dfrac{8\pi}{7}, \dfrac{9\pi}{7}, \dfrac{10\pi}{7}, \dfrac{11\pi}{7}, \dfrac{12\pi}{7}, \dfrac{13\pi}{7}[/latex]
  64. [latex]t=0, \dfrac{\pi}{2}, \pi, \dfrac{3\pi}{2}[/latex]
  65. [latex]x = 0[/latex]
  66. [latex]\theta = \dfrac{\pi}{48}, \dfrac{11\pi}{48}, \dfrac{13\pi}{48}, \dfrac{23\pi}{48}, \dfrac{25\pi}{48}, \dfrac{35\pi}{48}, \dfrac{37\pi}{48}, \dfrac{47\pi}{48}, \dfrac{49\pi}{48}, \dfrac{59\pi}{48}, \dfrac{61\pi}{48}, \dfrac{71\pi}{48}, \dfrac{73\pi}{48}, \dfrac{83\pi}{48}, \dfrac{85\pi}{48}, \dfrac{95\pi}{48}[/latex]
  67. [latex]t = 0, \dfrac{\pi}{2}[/latex]
  68. [latex]x = \dfrac{\pi}{2}, \dfrac{11\pi}{6}[/latex]
  69. [latex]\theta = \dfrac{\pi}{12}, \dfrac{17\pi}{12}[/latex]
  70. [latex]t = 0, \pi, \dfrac{\pi}{3}, \dfrac{4\pi}{3}[/latex]
  71. [latex]x = \dfrac{17 \pi}{24}, \dfrac{41 \pi}{24}, \dfrac{23\pi}{24}, \dfrac{47\pi}{24}[/latex]
  72. [latex]\theta = \dfrac{\pi}{6}, \dfrac{5\pi}{18}, \dfrac{5\pi}{6}, \dfrac{17\pi}{18}, \dfrac{3\pi}{2}, \dfrac{29\pi}{18}[/latex]
  73. [latex]t = 0, \dfrac{\pi}{4}, \dfrac{\pi}{2}, \dfrac{3\pi}{4}, \pi, \dfrac{5\pi}{4}, \dfrac{3\pi}{2}, \dfrac{7\pi}{4}[/latex]
  74. [latex]x = 0, \dfrac{\pi}{3}, \dfrac{2\pi}{3}, \pi, \dfrac{4\pi}{3}, \dfrac{5\pi}{3}[/latex]
  75. [latex]\theta = 0, \dfrac{\pi}{8}, \dfrac{3\pi}{8}, \dfrac{5\pi}{8}, \dfrac{7\pi}{8}, \pi, \dfrac{9\pi}{8}, \dfrac{11\pi}{8}, \dfrac{13\pi}{8}, \dfrac{15\pi}{8}[/latex]
  76. [latex]t = \dfrac{\pi}{7}, \dfrac{\pi}{3}, \dfrac{3\pi}{7}, \dfrac{5\pi}{7}, \pi, \dfrac{9\pi}{7}, \dfrac{11\pi}{7}, \dfrac{5\pi}{3}, \dfrac{13\pi}{7}[/latex]
  77. [latex]x = 0, \dfrac{2\pi}{7}, \dfrac{4\pi}{7}, \dfrac{6\pi}{7}, \dfrac{8\pi}{7}, \dfrac{10\pi}{7}, \dfrac{12\pi}{7}, \dfrac{\pi}{5}, \dfrac{3\pi}{5}, \pi, \dfrac{7\pi}{5}, \dfrac{9\pi}{5}[/latex]
  78. [latex]x = \arcsin \left( \dfrac{-1 + \sqrt{5}}{2} \right) \approx 0.6662, \pi - \arcsin \left( \dfrac{-1 + \sqrt{5}}{2} \right) \approx 2.4754[/latex]
  79. [latex]x = -\frac{1}{2}[/latex]
  80. [latex]t=-1[/latex]
  81. [latex]x = \frac{2}{3}[/latex]
  82. [latex]t=-\frac{\sqrt{3}}{2}[/latex]
  83. [latex]x = 2\sqrt{2}[/latex]
  84. [latex]t = 6[/latex]
  85. [latex]x = \pm \frac{\sqrt{3}}{2}[/latex]
  86. [latex]t = \frac{1}{2}[/latex]
  87. [latex]x = -1,0[/latex]
  88. [latex]t = -\sqrt{3}[/latex]
  89. Answer May Vary
  90. .
    1. [latex]k = 5 \, \frac{\text{lbs.}}{\text{ft.}}[/latex] and [latex]m = \frac{5}{16}[/latex] slugs
    2. [latex]x(t) = \sin\left(4t + \frac{\pi}{2}\right)[/latex]. The object first passes through the equilibrium point when [latex]t = \frac{\pi}{8} \approx 0.39[/latex] seconds after the motion starts. At this time, the object is heading upwards.
    3. [latex]x(t) = \frac{\sqrt{2}}{2} \sin\left(4t + \frac{7\pi}{4}\right)[/latex]. The object passes through the equilibrium point heading downwards for the third time when [latex]t = \frac{17\pi}{16} \approx 3.34[/latex] seconds.

Section 8.4 Answers

  1. [latex]\alpha = 13^{\circ}, \; \beta = 17^{\circ}, \; \gamma = 150^{\circ}, \; a = 5, \; b \approx 6.50, \; c \approx 11.11[/latex]
  2. [latex]\alpha = 73.2^{\circ}, \; \beta = 54.1^{\circ}, \; \gamma = 52.7^{\circ}, \; a = 117, \; b \approx 99.00, \; c \approx 97.22[/latex]
  3. Information does not produce a triangle
  4. [latex]\alpha = 95^{\circ}, \; \beta = 62^{\circ}, \; \gamma = 23^{\circ}, \;  a = 33.33, \; b \approx 29.54, \; c \approx 13.07[/latex]
  5. Information does not produce a triangle
  6. [latex]\alpha = 117^{\circ}, \; \beta \approx 56.3^{\circ}, \; \gamma \approx 6.7^{\circ}, \; a = 45, \; b = 42, \; c \approx 5.89[/latex]
  7. [latex]\alpha = 68.7^{\circ}, \; \beta \approx 76.9^{\circ}, \; \gamma \approx 34.4^{\circ}, \; a = 88, \; b = 92, \; c \approx 53.36[/latex]
    [latex]\alpha = 68.7^{\circ}, \; \beta \approx 103.1^{\circ}, \; \gamma \approx 8.2^{\circ}, \; a = 88, \; b = 92, \; c \approx 13.47[/latex]
  8. [latex]\alpha = 42^{\circ}, \; \beta \approx 67.66^{\circ}, \; \gamma \approx 70.34^{\circ}, \; a = 17, \; b = 23.5, \; c \approx 23.93[/latex]
    [latex]\alpha = 42^{\circ}, \; \beta \approx 112.34^{\circ}, \; \gamma \approx 25.66^{\circ}, \; a = 17, \; b = 23.5, \; c \approx 11.00[/latex]
  9. Information does not produce a triangle
  10. [latex]\alpha = 30^{\circ}, \; \beta = 90^{\circ}, \; \gamma = 60^{\circ}, \; a = 7, \; b = 14, \; c = 7\sqrt{3}[/latex]
  11. [latex]\alpha = 42^{\circ}, \; \beta \approx 23.78^{\circ}, \; \gamma \approx 114.22^{\circ}, \; a = 39, \; b = 23.5, \; c \approx 53.15[/latex]
  12. [latex]\alpha = 53^{\circ}, \; \beta = 74^{\circ}, \; \gamma = 53^{\circ}, \; a = 28.01, \; b \approx 33.71, \; c = 28.01[/latex]
  13. [latex]\alpha = 6^{\circ}, \; \beta \approx 169.43^{\circ}, \; \gamma \approx 4.57^{\circ}, \; a = 57, \; b = 100, \; c \approx 43.45[/latex]
    [latex]\alpha = 6^{\circ}, \; \beta \approx 10.57^{\circ}, \; \gamma \approx 163.43^{\circ}, \; a = 57, \; b = 100 , \; c \approx 155.51[/latex]
  14. [latex]\alpha \approx 78.59^{\circ}, \; \beta \approx 26.81^{\circ}, \; \gamma = 74.6^{\circ}, \; a = 3.05, \; b \approx 1.40, \; c = 3[/latex]
    [latex]\alpha \approx 101.41^{\circ}, \; \beta \approx 3.99^{\circ}, \; \gamma = 74.6^{\circ}, \; a = 3.05, \; b \approx 0.217, \; c = 3[/latex]
  15. [latex]\alpha \approx 28.61^{\circ}, \; \beta = 102^{\circ}, \; \gamma \approx 49.39^{\circ}, \; a \approx 8.20, \; b = 16.75, \; c = 13[/latex]
  16. Information does not produce a triangle
  17. [latex]\alpha = 43^{\circ}, \; \beta = 102^{\circ}, \; \gamma = 35^{\circ}, \; a \approx 11.68, \; b = 16.75, \; c \approx 9.82[/latex]
  18. [latex]\alpha = 66.92^{\circ}, \; \beta = 29.13^{\circ}, \; \gamma = 83.95^{\circ}, \; a \approx 593.69, \; b = 314.15, \; c \approx 641.75[/latex]
  19. Information does not produce a triangle
  20. $\alpha = 50^{\circ}, \; \beta \approx 22.52^{\circ}, \; \gamma \approx 107.48^{\circ}, \;
    a = 25, \; b = 12.5, \; c \approx 31.13$
  21.  The area of the triangle from Exercise 1 is about 8.1 square units.
    The area of the triangle from Exercise 12 is about 377.1 square units.
    The area of the triangle from Exercise 20 is about 149 square units.
  22. [latex]\arctan\left(\frac{7}{100}\right) \approx 0.699[/latex] radians, which is equivalent to [latex]4.004^{\circ}[/latex]
  23. About 17\%
  24. About 53 feet
  25. .
    1. [latex]\theta = 180^{\circ}[/latex]
    2. [latex]\theta = 353^{\circ}[/latex]
    3. [latex]\theta = 84.5^{\circ}[/latex]
    4. [latex]\theta = 270^{\circ}[/latex] [latex]\theta = 121.25^{\circ}[/latex]
    5. [latex]\theta = 45^{\circ}[/latex]
    6. [latex]\theta = 225^{\circ}[/latex]
  26. The Colonel is about 3193 feet from the campfire.
    Sarge is about 2525 feet to the campfire.
  27. The distance from the Muffin Ridge Observatory to Sasquach Point is about 7.12 miles.
    The distance from Sasquatch Point to the Chupacabra Trailhead is about 2.46 miles.
  28. The SS Bigfoot is about 4.1 miles from the flare.
    The HMS Sasquatch is about 2.9 miles from the flare.
  29. Jeff is about 371 feet from the nest.
  30. She is about 3.02 miles from the lodge
  31. The boat is about 25.1 miles from the second tower.
  32. The UFO is hovering about 9539 feet above the ground.
  33. The gargoyle is about 44 feet from the observer on the upper floor.
    The gargoyle is about 27 feet from the observer on the lower floor.
    The gargoyle is about 25 feet from the other building.

Section 8.5 Answers

  1. [latex]\alpha \approx 35.54^{\circ}, \; \beta \approx 85.16^{\circ}, \; \gamma = 59.3^{\circ}, \; a = 7, \; b = 12, \; c \approx 10.36[/latex]
  2. [latex]\alpha = 104^{\circ}, \; \beta \approx 29.40^{\circ}, \; \gamma \approx 46.60^{\circ}, \; a \approx 49.41, \; b = 25, \; c = 37[/latex]
  3. [latex]\alpha \approx 85.90^{\circ}, \; \beta = 8.2^{\circ}, \; \gamma \approx 85.90^{\circ}, \; a = 153, \; b \approx 21.88, \; c = 153[/latex]
  4. [latex]\alpha \approx 36.87^{\circ}, \; \beta \approx 53.13^{\circ} , \; \gamma = 90^{\circ}, \; a = 3, \; b = 4 , \; c = 5[/latex]
  5. [latex]\alpha = 120^{\circ}, \; \beta \approx 25.28^{\circ}, \; \gamma \approx 34.72^{\circ}, \; a = \sqrt{37} , \; b = 3, \; c = 4[/latex]
  6. [latex]\alpha \approx 32.31^{\circ}, \; \beta \approx 49.58^{\circ}, \; \gamma \approx 98.21^{\circ}, \; a = 7, \; b = 10, \; c = 13[/latex]
  7. Information does not produce a triangle
  8. [latex]\alpha \approx 83.05^{\circ}, \; \beta \approx 87.81^{\circ}, \; \gamma \approx 9.14^{\circ}, \; a = 300, \; b = 302, \; c = 48[/latex]
  9. [latex]\alpha = 60^{\circ}, \; \beta = 60^{\circ}, \; \gamma = 60^{\circ}, \; a = 5, \; b = 5, \; c = 5[/latex]
  10. [latex]\alpha \approx 22.62^{\circ}, \; \beta \approx 67.38^{\circ}, \; \gamma = 90^{\circ}, \; a = 5, \; b = 12 , \; c = 13[/latex]
  11. [latex]\alpha = 63^{\circ}, \; \beta \approx 98.11^{\circ}, \; \gamma \approx 18.89^{\circ} , \; a = 18, \; b = 20, \; c \approx 6.54[/latex]
    [latex]\alpha = 63^{\circ}, \; \beta \approx 81.89^{\circ}, \; \gamma \approx 35.11^{\circ}, \; a = 18, \; b = 20, \; c \approx 11.62[/latex]
  12. [latex]\alpha \approx 55.30^{\circ}, \; \beta \approx 89.40^{\circ}, \; \gamma \approx 35.30^{\circ}, \; a = 37, \; b = 45, \; c = 26[/latex]
  13. Information does not produce a triangle
  14. [latex]\alpha = 63^{\circ}, \; \beta \approx 54.1^{\circ}, \; \gamma \approx 62.9^{\circ}, \;  a = 22, \; b = 20, \; c \approx 21.98[/latex]
  15. [latex]\alpha = 42^{\circ}, \;   \beta \approx 89.23^{\circ}, \;   \gamma \approx 48.77^{\circ}, \; a \approx 78.30, \; b = 117, \; c = 88[/latex]
  16. [latex]\alpha \approx 3^{\circ}, \;  \beta = 7^{\circ}, \;   \gamma = 170^{\circ}, \;  a \approx 29.72, \; b \approx 69.2, \; c = 98.6[/latex]
  17. The area of the triangle given in Exercise 6 is [latex]\sqrt{1200} = 20\sqrt{3} \approx 34.64[/latex] square units.
    The area of the triangle given in Exercise 8 is [latex]\sqrt{51764375} \approx 7194.75[/latex] square units.
    The area of the triangle given in Exercise 10 is exactly 30 square units.
  18. The distance between the ends of the hands at four o’clock is about 8.26 inches.
  19. The diameter of the crater is about 5.22 miles.
  20. About 313 miles
  21. N[latex]31.8^{\circ}[/latex]W
  22. She is about 3.92 miles from the lodge and her bearing to the lodge is N[latex]37^{\circ}[/latex]E
  23. It is about 4.50 miles from port and its heading to port is S[latex]47^{\circ}[/latex]W.
  24. It is about 229.61 miles from the island and the captain should set a course of N[latex]16.4^{\circ}[/latex]E to reach the island.
  25. The fires are about 17456 feet apart. (Try to avoid rounding errors.)

 

License

Icon for the Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License

Functions, Trigonometry, and Systems of Equations (Second Edition) Copyright © 2025 by Texas A&M Mathematics Department is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License, except where otherwise noted.

Share This Book