9.1.2: Homework

Concept Check

1. We showed that \(\sin \left(\theta\right) = \pm \sqrt{1 - \cos^2 \left(\theta\right)}\) for all \(\theta \). Give an example of an angle \(\theta \) such that \(\sin\left(\theta\right) = -\sqrt{1 - \cos^2\left(\theta\right)} \).

2. We showed that \(\cos\left(\theta\right) = \pm \sqrt{1 - \sin^2\left(\theta\right)} \) for all \(\theta \). Give an example of an angle \(\theta \) such that \(\cos\left(\theta\right) = -\sqrt{1 - \sin^2\left(\theta\right)} \).

3. A student claims that \(\cos(\theta) + \sin(\theta) = 1\) is an identity because \(\cos(0) + \sin(0) = 1 + 0 = 0\). How would you respond to this student?

Proofs

For the following exercises, prove the identity. Assume all quantities are defined.

4. \( \tan\left( \theta \right) + \tan\left( -\theta \right) = 0\)

5. \( \cos^2\left( \theta \right) - \sin^2\left( \theta \right) = 2\cos^2\left( \theta \right) - 1 \)

6. \( \cos^2\left( \theta \right) - \sin^2\left( \theta \right) = 1 - 2\sin^2\left( \theta \right) \)

7. \( \dfrac{\sin^2\left( \theta \right)}{1 - \sin^2\left( \theta \right)} = \tan^2\left( \theta \right) \)

8. \( \sin^4\left( \theta \right) - \cos^4\left( \theta \right) = \sin^2\left( \theta \right) - \cos^2\left( \theta \right) \)

9. \( \sin^2\left( x \right) + \tan^2\left( x \right) + \cos^2\left( x \right) = \sec^2\left( x \right) \)

10. \( \cos^4\left( \theta \right) - \sin^4\left( \theta \right) = 1 - 2\sin^2\left( \theta \right) \)

11. \(\dfrac{\sin(x) + \cos(x)}{\cos(x)} = \tan(x) + 1\)

12. \( \dfrac{\cos^2\left( \theta \right)}{1 + \sin\left( \theta \right)} = 1 - \sin\left( \theta \right) \)

13. \(\dfrac{\sin^2(x)}{1 + \cos(x)} = 1 - \cos(x)\)

14. \(\dfrac{\cos(x)}{1 + \sin(x)} = \sec(x) - \tan(x)\)

15. \(\dfrac{\sin(x) - \cos(x)}{\sin(x) + \cos(x)} = \dfrac{\tan(x) - 1}{1 + \tan(x)}\)

16. \( \dfrac{1 - \tan\left( \theta \right)}{1 + \tan\left( \theta \right)} = \dfrac{\cot\left( \theta \right) - 1}{\cot\left( \theta \right) + 1} \)

17. \(\dfrac{1}{\tan(x) + \cot(x)} = \sin(x) \cdot \cos(x)\)

18. \(\dfrac{\tan(x) + \sec(x) - 1}{\tan(x) - \sec(x) + 1} = \dfrac{1 + \sin(x)}{\cos(x)}\)

19. \(\dfrac{1 + \sin(x)}{\cos(x)} = \sec(x) + \tan(x)\)

20. \(\dfrac{\sin(x) - \cos(x)}{\cos(-x) - \sin(-x)} = \dfrac{-1 - \tan(-x)}{1 + \tan(x)}\)

21. \(\dfrac{\sin(-x)}{1 + \cos(x)} = \dfrac{\cos(-x) - 1}{\sin(x)}\)

22. \(\dfrac{\sec(x) - 1}{\sec(x) + 1} = \dfrac{\sin^2(x)}{(\cos(x) + 1)^2}\)

23. \(\dfrac{\sec(x) - 1 - \tan(-x)}{\tan(x) - \sec(-x) + 1} = \dfrac{1 - \sin(-x)}{\cos(x)}\)

24. \(\dfrac{\cos(-x)}{1 - \sin(-x)} = \sec(-x) + \tan(-x)\)

25. \(\dfrac{1 - \sin(-x)}{\cos(x)} = \sec(-x) - \tan(-x)\)

26. \(\sin(3\pi - 2\theta) = -\sin(2\theta - 3\pi)\)

27. \(\dfrac{\cos(x)}{1 - \sin(x)} = \sec(x) + \tan(x)\)

28. \(\dfrac{1 + \cos\left( \frac{\pi}{2} - x \right)}{\cos(x)} = \csc\left( \frac{\pi}{2} - x \right) + \tan(x)\)

29. \(\dfrac{1 - \cos(x)}{\cos(x)} = \dfrac{\sin^2(x)}{\cos(x)(1 + \cos(x))}\)

30. \(\dfrac{\sin^2(x)}{\cos(x)} = \tan(x) \cdot \sin(x)\)

31. \(\dfrac{\sin(-x) - \cos(x)}{\cos(x)} = \tan(-x) - 1\)

32. \( \dfrac{1 - 2\cos^2\left( \theta \right)}{\sin\left( \theta \right) \cos\left( \theta \right)} = \tan\left( \theta \right) - \cot\left( \theta \right)\)

33. \(\cos \left( -\dfrac{\pi}{4} - 5t \right) = \cos \left( 5t + \dfrac{\pi}{4} \right)\)

34. \(\dfrac{\cos\left( \frac{\pi}{2} - x \right)}{\cos(x) + 1} = \dfrac{1 - \sin\left( \frac{\pi}{2} - x \right)}{\sin(x)}\)

35. \( \dfrac{1 - \tan^2\left( \theta \right)}{1 - \cot^2\left( \theta \right)} = 1 - \sec^2\left( \theta \right) \)

36. \(\dfrac{\cos(-x)}{1 + \sin(-x)} = \sec(x) - \tan(-x)\)

37. \(\dfrac{1 + \cos(x)}{\sin(x)} = \csc(x) + \cot(x)\)

38. \(\dfrac{1 - \sin^2(x)}{\cos(x)} = \cos(x)\)

39. \(\dfrac{1 - \sin(x)}{1 + \sin(x)} = \dfrac{\sec(x) - \tan(x)}{\sec(x) + \tan(x)}\)

40. \(\dfrac{\sec(x) - 1}{\sec(x) + 1} = \dfrac{1 - \cos(x)}{1 + \cos(x)}\)

41. \(\dfrac{1 + \sin(x)}{\cos(x)} = \dfrac{\cos(x)}{1 - \sin(x)}\)

42. \(\tan(-t^{2} + 1) = -\tan(t^{2} - 1)\)

43. \(\dfrac{1 - \cos\left( \frac{\pi}{2} - x \right)}{\cos(x)} = \csc\left( \frac{\pi}{2} - x \right) - \tan(x)\)

44. \(\dfrac{\sin(x) - \sin\left(\dfrac{\pi}{2} - x\right)}{\sin(x) + \cos(x)} = \dfrac{\cot\left(\dfrac{\pi}{2} - x\right) - 1}{1 + \tan(x)}\)

45. \(\dfrac{\cot\left( \frac{\pi}{2} - B \right) + \sec(B) - 1}{\tan(B) - \csc\left( \frac{\pi}{2} - B \right) + 1} = \dfrac{1 - \sin(-B)}{\cos(B)}\)

46. \(\dfrac{\sin(\phi)}{1 + \cos(\phi)} = \dfrac{1 - \cos(\phi)}{\sin(\phi)}\)

47. \(\dfrac{\cos^2(\theta)}{1 - \sin(\theta)} = 1 + \sin(\theta)\)

48. \(\csc(-\theta - 5) = -\csc(\theta + 5)\)

49. \(\dfrac{\sin\left(\dfrac{\pi}{2} - \alpha\right)}{1 + \sin(\alpha)} = \csc\left(\dfrac{\pi}{2} - \alpha\right) - \tan(\alpha)\)

50. \(\dfrac{1 + \sin(-\theta)}{1 + \sin(\theta)} = \dfrac{\sec(-\theta) + \tan(-\theta)}{\sec(-\theta) + \tan(\theta)}\)

51. \(\dfrac{1 + \cos(-A)}{\sin(-A)} = -\csc(A) + \cot(-A)\)

52. \(\dfrac{\cos(-\phi) - 1}{\cos(\phi)} = \dfrac{\sin^2(-\phi)}{\cos(\phi)(1 + \cos(\phi))}\)

53. \(\dfrac{\sin\left( \frac{\pi}{2} - \theta \right)}{1 - \sin(\theta)} = \csc\left( \frac{\pi}{2} - \theta \right) + \cot\left( \frac{\pi}{2} - \theta \right)\)

54. \(\dfrac{\sec(-\alpha) - 1}{\sec(\alpha) + 1} = \dfrac{\sin^2(-\alpha)}{(\cos(-\alpha) + 1)^2}\)

55. \(\sec(-6t) = \sec(6t)\)

56. \(-\dfrac{\cos^2(-A)}{1 + \sin(-A)} = -1 + \sin(-A)\)

57. \( \dfrac{\tan\left( \theta \right) + \tan\left( \phi \right)}{\cot\left( \theta \right) + \cot\left( \phi \right)} = \tan\left( \theta \right) \tan\left( \phi \right) \)

58. \(\dfrac{\sin^2(-\alpha)}{\cos(-\alpha)} = -\tan(\alpha) \cdot \sin(-\alpha)\)

59. \(\dfrac{\sec(y) - 1}{\sec(y) + 1} = \dfrac{1 - \cos(-y)}{1 + \cos(-y)}\)

60. \(\dfrac{\sin^2\left( \frac{\pi}{2} - A \right)}{1 - \sin(A)} = \dfrac{\cos^2\left( A \right)}{1 - \cos\left(\frac{\pi}{2} - A\right)}\)

61. \(\dfrac{1 + \sin\left( \frac{\pi}{2} - B \right)}{\sin(B)} = \sec\left( \frac{\pi}{2} - B \right) + \cot(B)\)

62. \(\dfrac{\sin\left( \frac{\pi}{2} - \phi \right) - 1}{\sin\left( \frac{\pi}{2} - \phi \right)} = \dfrac{\sin^2(\phi)}{\sin\left( \frac{\pi}{2} - \phi \right)\left(1 + \sin\left( \frac{\pi}{2} - \phi \right)\right)}\)

63. \(\cot(9 - 7\theta) = -\cot(7\theta - 9)\)

64. \(\dfrac{\csc\left( \frac{\pi}{2} - B \right) - 1}{\sec(B) + 1} = \dfrac{\cos^2\left( \frac{\pi}{2} - B \right)}{\left(\sin\left( \frac{\pi}{2} - B \right) + 1\right)^2}\)

65. \(\dfrac{1 - \cos(\alpha)}{\sin^2(\alpha)} = \dfrac{1}{1 + \cos(\alpha)}\)

Challenge Problems

66. Suppose that you are given a system of two equations of the following form:\[\begin{array}{rcl}
    A \cos\left(\phi\right) & = & B \nu_1 - B\nu_2 \cos\left(\theta\right) \\[6pt] A\sin\left(\phi\right) & = & B \nu_2 \sin\left(\theta\right) \\[6pt] \end{array} \nonumber \]Show that\[A^2 = B^2 \left( \nu_1^2 + \nu_2^2 - 2\nu_1 \nu_2 \cos\left(\theta\right) \right).\nonumber \]

67. Suppose that a point with coordinates\[\left(x,y\right) = \left(a \left(\cos\left(\psi\right) - \epsilon\right), a\sqrt{1 - \epsilon^2} \sin\left(\psi\right)\right) \nonumber \]is a distance \(r>0 \) from the origin, where \(a>0 \) and \(0 < \epsilon < 1 \). Use \(r^2 = x^2 + y^2\) to show that \(r = a\left(1 - \epsilon \cos\left(\psi\right) \right)\).
    (Note: These coordinates arise in the study of elliptical orbits of planets.)

68. Show that each trigonometric function can be written solely in terms of the sine function.