Simplifying Trigonometric Expressions with Identities

3. When [latex]\cos t=0[/latex], then [latex]\sec t=\frac{1}{0}[/latex], which is undefined.

5. [latex]\sin x[/latex]

7. [latex]\sec x[/latex]

9. [latex]\csc t[/latex]

11. [latex]-1[/latex]

13. [latex]{\sec }^{2}x[/latex]

15. [latex]{\sin }^{2}x+1[/latex]

29. Answers will vary. Sample proof:
[latex]\cos x-{\cos }^{3}x=\cos x\left(1-{\cos }^{2}x\right)[/latex]
[latex]=\cos x{\sin }^{2}x[/latex]

31. Answers will vary. Sample proof:

[latex]\frac{1+{\sin }^{2}x}{{\cos }^{2}x}=\frac{1}{{\cos }^{2}x}+\frac{{\sin }^{2}x}{{\cos }^{2}x}={\sec }^{2}x+{\tan }^{2}x={\tan }^{2}x+1+{\tan }^{2}x=1+2{\tan }^{2}x[/latex]

33. Answers will vary. Sample proof:

[latex]{\cos }^{2}x-{\tan }^{2}x=1-{\sin }^{2}x-\left({\sec }^{2}x - 1\right)=1-{\sin }^{2}x-{\sec }^{2}x+1=2-{\sin }^{2}x-{\sec }^{2}x[/latex]

39. Proved with negative and Pythagorean identities

Sum and Difference Identities

1. The cofunction identities apply to complementary angles. Viewing the two acute angles of a right triangle, if one of those angles measures [latex]x[/latex], the second angle measures [latex]\frac{\pi }{2}-x[/latex]. Then [latex]\sin x=\cos \left(\frac{\pi }{2}-x\right)[/latex]. The same holds for the other cofunction identities. The key is that the angles are complementary.

5. [latex]\frac{\sqrt{2}+\sqrt{6}}{4}[/latex]

7. [latex]\frac{\sqrt{6}-\sqrt{2}}{4}[/latex]

9. [latex]-2-\sqrt{3}[/latex]

11. [latex]-\frac{\sqrt{2}}{2}\sin x-\frac{\sqrt{2}}{2}\cos x[/latex]

13. [latex]-\frac{1}{2}\cos x-\frac{\sqrt{3}}{2}\sin x[/latex]

15. [latex]\csc \theta[/latex]

17. [latex]\cot x[/latex]

19. [latex]\tan \left(\frac{x}{10}\right)[/latex]

21. [latex]\sin \left(a-b\right)=\left(\frac{4}{5}\right)\left(\frac{1}{3}\right)-\left(\frac{3}{5}\right)\left(\frac{2\sqrt{2}}{3}\right)=\frac{4 - 6\sqrt{2}}{15}[/latex]
[latex]\cos \left(a+b\right)=\left(\frac{3}{5}\right)\left(\frac{1}{3}\right)-\left(\frac{4}{5}\right)\left(\frac{2\sqrt{2}}{3}\right)=\frac{3 - 8\sqrt{2}}{15}[/latex]

23. [latex]\frac{\sqrt{2}-\sqrt{6}}{4}[/latex]

25. [latex]\sin x[/latex]

27. [latex]\cot \left(\frac{\pi }{6}-x\right)[/latex]

31. [latex]\frac{\sin x}{\sqrt{2}}+\frac{\cos x}{\sqrt{2}}[/latex]

43. [latex]-\frac{\sqrt{3}-1}{2\sqrt{2}},\text{ or }-0.2588[/latex]

45. [latex]\frac{1+\sqrt{3}}{2\sqrt{2}}[/latex], or 0.9659

47. [latex]\begin{array}{c}\tan \left(x+\frac{\pi }{4}\right)=\\ \frac{\tan x+\tan \left(\frac{\pi }{4}\right)}{1-\tan x\tan \left(\frac{\pi }{4}\right)}=\\ \frac{\tan x+1}{1-\tan x\left(1\right)}=\frac{\tan x+1}{1-\tan x}\end{array}[/latex]

 

49. [latex]\begin{array}{c}\frac{\cos \left(a+b\right)}{\cos a\cos b}=\\ \frac{\cos a\cos b}{\cos a\cos b}-\frac{\sin a\sin b}{\cos a\cos b}=1-\tan a\tan b\end{array}[/latex]

51. [latex]\begin{array}{c}\frac{\cos \left(x+h\right)-\cos x}{h}=\\ \frac{\cos x\mathrm{cosh}-\sin x\mathrm{sinh}-\cos x}{h}=\\ \frac{\cos x\left(\mathrm{cosh}-1\right)-\sin x\mathrm{sinh}}{h}=\cos x\frac{\cos h - 1}{h}-\sin x\frac{\sin h}{h}\end{array}[/latex]

53. True

55. True. Note that [latex]\sin \left(\alpha +\beta \right)=\sin \left(\pi -\gamma \right)[/latex] and expand the right hand side.

Double-Angle, Half-Angle, and Reduction Formulas

1. Use the Pythagorean identities and isolate the squared term.

5. a) [latex]\frac{3\sqrt{7}}{32}[/latex] b) [latex]\frac{31}{32}[/latex] c) [latex]\frac{3\sqrt{7}}{31}[/latex]

7. a) [latex]\frac{\sqrt{3}}{2}[/latex] b) [latex]-\frac{1}{2}[/latex] c) [latex]-\sqrt{3}[/latex]

15. [latex]\frac{\sqrt{2-\sqrt{3}}}{2}[/latex]

17. [latex]2+\sqrt{3}[/latex]

19. [latex]-1-\sqrt{2}[/latex]

21. a) [latex]\frac{3\sqrt{13}}{13}[/latex] b) [latex]-\frac{2\sqrt{13}}{13}[/latex] c) [latex]-\frac{3}{2}[/latex]

23. a) [latex]\frac{\sqrt{10}}{4}[/latex] b) [latex]\frac{\sqrt{6}}{4}[/latex] c) [latex]\frac{\sqrt{15}}{3}[/latex]

25. [latex]\frac{120}{169},-\frac{119}{169},-\frac{120}{119}[/latex]

27. [latex]\frac{2\sqrt{13}}{13},\frac{3\sqrt{13}}{13},\frac{2}{3}[/latex]

29. [latex]\cos \left({74}^{\circ }\right)[/latex]

31. [latex]\cos \left(18x\right)[/latex]

33. [latex]3\sin \left(10x\right)[/latex]

35. [latex]-2\sin \left(-x\right)\cos \left(-x\right)=-2\left(-\sin \left(x\right)\cos \left(x\right)\right)=\sin \left(2x\right)[/latex]

37. [latex]\begin{array}{l}\frac{\sin \left(2\theta \right)}{1+\cos \left(2\theta \right)}{\tan }^{2}\theta =\frac{2\sin \left(\theta \right)\cos \left(\theta \right)}{1+{\cos }^{2}\theta -{\sin }^{2}\theta }{\tan }^{2}\theta =\\ \frac{2\sin \left(\theta \right)\cos \left(\theta \right)}{2{\cos }^{2}\theta }{\tan }^{2}\theta =\frac{\sin \left(\theta \right)}{\cos \theta }{\tan }^{2}\theta =\\ \cot \left(\theta \right){\tan }^{2}\theta =\tan \theta \end{array}[/latex]

39. [latex]\frac{1+\cos \left(12x\right)}{2}[/latex]

41. [latex]\frac{3+\cos \left(12x\right)-4\cos \left(6x\right)}{8}[/latex]

43. [latex]\frac{2+\cos \left(2x\right)-2\cos \left(4x\right)-\cos \left(6x\right)}{32}[/latex]

45. [latex]\frac{3+\cos \left(4x\right)-4\cos \left(2x\right)}{3+\cos \left(4x\right)+4\cos \left(2x\right)}[/latex]

47. [latex]\frac{1-\cos \left(4x\right)}{8}[/latex]

49. [latex]\frac{3+\cos \left(4x\right)-4\cos \left(2x\right)}{4\left(\cos \left(2x\right)+1\right)}[/latex]

51. [latex]\frac{\left(1+\cos \left(4x\right)\right)\sin x}{2}[/latex]

55. [latex]\frac{2\tan x}{1+{\tan }^{2}x}=\frac{\frac{2\sin x}{\cos x}}{1+\frac{{\sin }^{2}x}{{\cos }^{2}x}}=\frac{\frac{2\sin x}{\cos x}}{\frac{{\cos }^{2}x+{\sin }^{2}x}{{\cos }^{2}x}}=[/latex]
[latex]\frac{2\sin x}{\cos x}.\frac{{\cos }^{2}x}{1}=2\sin x\cos x=\sin \left(2x\right)[/latex]

57. [latex]\frac{2\sin x\cos x}{2{\cos }^{2}x - 1}=\frac{\sin \left(2x\right)}{\cos \left(2x\right)}=\tan \left(2x\right)[/latex]

59. [latex]\begin{array}{l}\sin \left(x+2x\right)=\sin x\cos \left(2x\right)+\sin \left(2x\right)\cos x\hfill \\ =\sin x\left({\cos }^{2}x-{\sin }^{2}x\right)+2\sin x\cos x\cos x\hfill \\ =\sin x{\cos }^{2}x-{\sin }^{3}x+2\sin x{\cos }^{2}x\hfill \\ =3\sin x{\cos }^{2}x-{\sin }^{3}x\hfill \end{array}[/latex]

61. [latex]\begin{array}{l}\frac{1+\cos \left(2t\right)}{\sin \left(2t\right)-\cos t}=\frac{1+2{\cos }^{2}t - 1}{2\sin t\cos t-\cos t}\hfill \\ =\frac{2{\cos }^{2}t}{\cos t\left(2\sin t - 1\right)}\hfill \\ =\frac{2\cos t}{2\sin t - 1}\hfill \end{array}[/latex]

Sum-to-Product and Product-to-Sum Formulas

5. [latex]8\left(\cos \left(5x\right)-\cos \left(27x\right)\right)[/latex]

7. [latex]\sin \left(2x\right)+\sin \left(8x\right)[/latex]

9. [latex]\frac{1}{2}\left(\cos \left(6x\right)-\cos \left(4x\right)\right)[/latex]

11. [latex]2\cos \left(5t\right)\cos t[/latex]

13. [latex]2\cos \left(7x\right)[/latex]

15. [latex]2\cos \left(6x\right)\cos \left(3x\right)[/latex]

17. [latex]\frac{1}{4}\left(1+\sqrt{3}\right)[/latex]

19. [latex]\frac{1}{4}\left(\sqrt{3}-2\right)[/latex]

21. [latex]\frac{1}{4}\left(\sqrt{3}-1\right)[/latex]

23. [latex]\cos \left(80^\circ \right)-\cos \left(120^\circ \right)[/latex]

25. [latex]\frac{1}{2}\left(\sin \left(221^\circ \right)+\sin \left(205^\circ \right)\right)[/latex]

27. [latex]\sqrt{2}\cos \left(31^\circ \right)[/latex]

29. [latex]2\cos \left(66.5^\circ \right)\sin \left(34.5^\circ \right)[/latex]

31. [latex]2\sin \left(-1.5^\circ \right)\cos \left(0.5^\circ \right)[/latex]

33. [latex]{2}\sin \left({7x}\right){-2}\sin{ x}={ 2}\sin \left({4x}+{ 3x }\right)-{ 2 }\sin\left({4x } - { 3x }\right)=\\ {2}\left(\sin\left({ 4x }\right)\cos\left({ 3x }\right)+\sin\left({ 3x }\right)\cos\left({ 4x }\right)\right)-{ 2 }\left(\sin\left({ 4x }\right)\cos\left({ 3x }\right)-\sin \left({ 3x }\right)\cos\left({ 4x }\right)\right)=\\{2}\sin\left({ 4x }\right)\cos\left({ 3x }\right)+{2}\sin\left({ 3x }\right)\cos\left({ 4x }\right)-{ 2 }\sin\left({ 4x }\right)\cos\left({ 3x }\right)+{ 2 }\sin\left({ 3x }\right)\cos\left({ 4x }\right)=\\{ 4 }\sin\left({ 3x }\right)\cos\left({ 4x }\right)\\[/latex]

 

35. [latex]\sin x+\sin \left(3x\right)=2\sin \left(\frac{4x}{2}\right)\cos \left(\frac{-2x}{2}\right)=[/latex]
[latex]2\sin \left(2x\right)\cos x=2\left(2\sin x\cos x\right)\cos x=[/latex]
[latex]4\sin x{\cos }^{2}x[/latex]

37. [latex]2\tan x\cos \left(3x\right)=\frac{2\sin x\cos \left(3x\right)}{\cos x}=\frac{2\left(.5\left(\sin \left(4x\right)-\sin \left(2x\right)\right)\right)}{\cos x}[/latex]
[latex]=\frac{1}{\cos x}\left(\sin \left(4x\right)-\sin \left(2x\right)\right)=\sec x\left(\sin \left(4x\right)-\sin \left(2x\right)\right)[/latex]

39. [latex]2\cos \left({35}^{\circ }\right)\cos \left({23}^{\circ }\right),\text{ 1}\text{.5081}[/latex]

41. [latex]-2\sin \left({33}^{\circ }\right)\sin \left({11}^{\circ }\right),\text{ }-0.2078[/latex]

43. [latex]\frac{1}{2}\left(\cos \left({99}^{\circ }\right)-\cos \left({71}^{\circ }\right)\right),\text{ }-0.2410[/latex]

49. [latex]\tan \left(3t\right)[/latex]

51. [latex]2\cos \left(2x\right)[/latex]

53. [latex]-\sin \left(14x\right)[/latex]

57. [latex]\frac{\cos \left(3x\right)+\cos x}{\cos \left(3x\right)-\cos x}=\frac{2\cos \left(2x\right)\cos x}{-2\sin \left(2x\right)\sin x}=-\cot \left(2x\right)\cot x[/latex]

59. [latex]\begin{array}{l}\frac{\cos \left(2y\right)-\cos \left(4y\right)}{\sin \left(2y\right)+\sin \left(4y\right)}=\frac{-2\sin \left(3y\right)\sin \left(-y\right)}{2\sin \left(3y\right)\cos y}=\\ \frac{2\sin \left(3y\right)\sin \left(y\right)}{2\sin \left(3y\right)\cos y}=\tan y\end{array}[/latex]

61. [latex]\begin{array}{l}\cos x-\cos \left(3x\right)=-2\sin \left(2x\right)\sin \left(-x\right)=\\ 2\left(2\sin x\cos x\right)\sin x=4{\sin }^{2}x\cos x\end{array}[/latex]

63. [latex]\tan \left(\frac{\pi }{4}-t\right)=\frac{\tan \left(\frac{\pi }{4}\right)-\tan t}{1+\tan \left(\frac{\pi }{4}\right)\tan \left(t\right)}=\frac{1-\tan t}{1+\tan t}[/latex]

Solving Trigonometric Equations

Solutions to Odd-Numbered Exercises

1. There will not always be solutions to trigonometric function equations. For a basic example, [latex]\cos \left(x\right)=-5[/latex].

5. [latex]\frac{\pi }{3},\frac{2\pi }{3}[/latex]

7. [latex]\frac{3\pi }{4},\frac{5\pi }{4}[/latex]

9. [latex]\frac{\pi }{4},\frac{5\pi }{4}[/latex]

11. [latex]\frac{\pi }{4},\frac{3\pi }{4},\frac{5\pi }{4},\frac{7\pi }{4}[/latex]

13. [latex]\frac{\pi }{4},\frac{7\pi }{4}[/latex]

15. [latex]\frac{7\pi }{6},\frac{11\pi }{6}[/latex]

17. [latex]\frac{\pi }{18},\frac{5\pi }{18},\frac{13\pi }{18},\frac{17\pi }{18},\frac{25\pi }{18},\frac{29\pi }{18}[/latex]

19. [latex]\frac{3\pi }{12},\frac{5\pi }{12},\frac{11\pi }{12},\frac{13\pi }{12},\frac{19\pi }{12},\frac{21\pi }{12}[/latex]

21. [latex]\frac{1}{6},\frac{5}{6},\frac{13}{6},\frac{17}{6},\frac{25}{6},\frac{29}{6},\frac{37}{6}[/latex]

23. [latex]0,\frac{\pi }{3},\pi ,\frac{5\pi }{3}[/latex]

25. [latex]\frac{\pi }{3},\pi ,\frac{5\pi }{3}[/latex]

27. [latex]\frac{\pi }{3},\frac{3\pi }{2},\frac{5\pi }{3}[/latex]

29. [latex]0,\pi[/latex]

31. [latex]\pi -{\sin }^{-1}\left(-\frac{1}{4}\right),\frac{7\pi }{6},\frac{11\pi }{6},2\pi +{\sin }^{-1}\left(-\frac{1}{4}\right)[/latex]

33. [latex]\frac{1}{3}\left({\sin }^{-1}\left(\frac{9}{10}\right)\right),\frac{\pi }{3}-\frac{1}{3}\left({\sin }^{-1}\left(\frac{9}{10}\right)\right),\frac{2\pi }{3}+\frac{1}{3}\left({\sin }^{-1}\left(\frac{9}{10}\right)\right),\pi -\frac{1}{3}\left({\sin }^{-1}\left(\frac{9}{10}\right)\right),\frac{4\pi }{3}+\frac{1}{3}\left({\sin }^{-1}\left(\frac{9}{10}\right)\right),\frac{5\pi }{3}-\frac{1}{3}\left({\sin }^{-1}\left(\frac{9}{10}\right)\right)[/latex]

35. [latex]0[/latex]

37. [latex]\frac{\pi }{6},\frac{5\pi }{6},\frac{7\pi }{6},\frac{11\pi }{6}[/latex]

39. [latex]\frac{3\pi }{2},\frac{\pi }{6},\frac{5\pi }{6}[/latex]

41. [latex]0,\frac{\pi }{3},\pi ,\frac{4\pi }{3}[/latex]

43. There are no solutions.

45. [latex]{\cos }^{-1}\left(\frac{1}{3}\left(1-\sqrt{7}\right)\right),2\pi -{\cos }^{-1}\left(\frac{1}{3}\left(1-\sqrt{7}\right)\right)[/latex]

47. [latex]{\tan }^{-1}\left(\frac{1}{2}\left(\sqrt{29}-5\right)\right),\pi +{\tan }^{-1}\left(\frac{1}{2}\left(-\sqrt{29}-5\right)\right),\pi +{\tan }^{-1}\left(\frac{1}{2}\left(\sqrt{29}-5\right)\right),2\pi +{\tan }^{-1}\left(\frac{1}{2}\left(-\sqrt{29}-5\right)\right)[/latex]

49. There are no solutions.

51. There are no solutions.

53. [latex]0,\frac{2\pi }{3},\frac{4\pi }{3}[/latex]

55. [latex]\frac{\pi }{4},\frac{3\pi }{4},\frac{5\pi }{4},\frac{7\pi }{4}[/latex]

57. [latex]{\sin }^{-1}\left(\frac{3}{5}\right),\frac{\pi }{2},\pi -{\sin }^{-1}\left(\frac{3}{5}\right),\frac{3\pi }{2}[/latex]

59. [latex]{\cos }^{-1}\left(-\frac{1}{4}\right),2\pi -{\cos }^{-1}\left(-\frac{1}{4}\right)[/latex]

61. [latex]\frac{\pi }{3},{\cos }^{-1}\left(-\frac{3}{4}\right),2\pi -{\cos }^{-1}\left(-\frac{3}{4}\right),\frac{5\pi }{3}[/latex]

63. [latex]{\cos }^{-1}\left(\frac{3}{4}\right),{\cos }^{-1}\left(-\frac{2}{3}\right),2\pi -{\cos }^{-1}\left(-\frac{2}{3}\right),2\pi -{\cos }^{-1}\left(\frac{3}{4}\right)[/latex]

65. [latex]0,\frac{\pi }{2},\pi ,\frac{3\pi }{2}[/latex]

67. [latex]\frac{\pi }{3},{\cos }^{-1}\left(-\frac{1}{4}\right),2\pi -{\cos }^{-1}\left(-\frac{1}{4}\right),\frac{5\pi }{3}[/latex]

69. There are no solutions.

71. [latex]\pi +{\tan }^{-1}\left(-2\right),\pi +{\tan }^{-1}\left(-\frac{3}{2}\right),2\pi +{\tan }^{-1}\left(-2\right),2\pi +{\tan }^{-1}\left(-\frac{3}{2}\right)[/latex]

73. [latex]2\pi k+0.2734,2\pi k+2.8682[/latex]

75. [latex]\pi k - 0.3277[/latex]

77. [latex]0.6694,1.8287,3.8110,4.9703[/latex]

79. [latex]1.0472,3.1416,5.2360[/latex]

81. [latex]0.5326,1.7648,3.6742,4.9064[/latex]

83. [latex]{\sin }^{-1}\left(\frac{1}{4}\right),\pi -{\sin }^{-1}\left(\frac{1}{4}\right),\frac{3\pi }{2}[/latex]

85. [latex]\frac{\pi }{2},\frac{3\pi }{2}[/latex]

87. There are no solutions.

89. [latex]0,\frac{\pi }{2},\pi ,\frac{3\pi }{2}[/latex]

91. There are no solutions.

93. [latex]{7.2}^{\circ }[/latex]

95. [latex]{5.7}^{\circ }[/latex]

97. [latex]{82.4}^{\circ }[/latex]

99. [latex]{31.0}^{\circ }[/latex]

101. [latex]{88.7}^{\circ }[/latex]

103. [latex]{59.0}^{\circ }[/latex]

105. [latex]{36.9}^{\circ }[/latex]

Modeling with Trigonometric Equations

1. Physical behavior should be periodic, or cyclical.

3. Since cumulative rainfall is always increasing, a sinusoidal function would not be ideal here.

5. [latex]y=-3\cos \left(\frac{\pi }{6}x\right)-1[/latex]

7. [latex]5\sin \left(2x\right)+2[/latex]

9. [latex]4\cos \left(\frac{x\pi }{2}\right)-3[/latex]

11. [latex]5 - 8\sin \left(\frac{x\pi }{2}\right)[/latex]

13. [latex]\tan \left(\frac{x\pi }{12}\right)[/latex]

17. 9 years from now

19. [latex]56^\circ \text{F}[/latex]

21. [latex]1.8024[/latex] hours

23. 4:30

25. From July 8 to October 23

27. From day 19 through day 40

29. Floods: July 24 through October 7. Droughts: February 4 through March 27

31. Amplitude: 11, period: [latex]\frac{1}{6}[/latex], frequency: 6 Hz

33. Amplitude: 5, period: [latex]\frac{1}{30}[/latex], frequency: 30 Hz

35. [latex]P\left(t\right)=-15\cos \left(\frac{\pi }{6}t\right)+650+\frac{55}{6}t[/latex]

37. [latex]P\left(t\right)=-40\cos \left(\frac{\pi }{6}t\right)+800{\left(1.04\right)}^{t}[/latex]

39. [latex]D\left(t\right)=7{\left(0.89\right)}^{t}\cos \left(40\pi t\right)[/latex]

41. [latex]D\left(t\right)=19{\left(0.9265\right)}^{t}\cos \left(26\pi t\right)[/latex]

43. [latex]20.1[/latex] years

45. 17.8 seconds

47. Spring 2 comes to rest first after 8.0 seconds.

49. 500 miles, at [latex]{90}^{\circ }[/latex]