Section 5.1 Solutions

1. All three functions, $F,G$, and $H$, are even.

This is because $F\left(-x\right)=\sin \left(-x\right)\sin \left(-x\right)=\left(-\sin x\right)\left(-\sin x\right)={\sin }^{2}x=F\left(x\right),G\left(-x\right)=\cos \left(-x\right)\cos \left(-x\right)=\cos x\cos x={\cos }^{2}x=G\left(x\right)$ and $H\left(-x\right)=\tan \left(-x\right)\tan \left(-x\right)=\left(-\tan x\right)\left(-\tan x\right)={\tan }^{2}x=H\left(x\right)$.

3. When $\cos t=0$, then $\sec t=\frac{1}{0}$, which is undefined.

5. $\sin x$

7. $\sec x$

9. $\csc t$

11. $-1$

13. ${\sec }^{2}x$

15. ${\sin }^{2}x+1$

17. $\frac{1}{\sin x}$

19. $\frac{1}{\cot x}$

21. $\tan x$

23. $-4\sec x\tan x$

25. $\pm \sqrt{\frac{1}{{\cot }^{2}x}+1}$

27. $\frac{\pm \sqrt{1-{\sin }^{2}x}}{\sin x}$

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

31. Answers will vary. Sample proof:

$\frac{1+{\sin }^{2}x}{{\cos }^{2}x}=\frac{1}{{\cos }^{2}x}+\frac{{\sin }^{2}x}{{\cos }^{2}x}={\sec }^{2}x+{\tan }^{2}x={\tan }^{2}x+1+{\tan }^{2}x=1+2{\tan }^{2}x$

33. Answers will vary. Sample proof:

${\cos }^{2}x-{\tan }^{2}x=1-{\sin }^{2}x-\left({\sec }^{2}x - 1\right)=1-{\sin }^{2}x-{\sec }^{2}x+1=2-{\sin }^{2}x-{\sec }^{2}x$

35. False

37. False

39. Proved with negative and Pythagorean identities

41. True

$3{\sin }^{2}\theta +4{\cos }^{2}\theta =3{\sin }^{2}\theta +3{\cos }^{2}\theta +{\cos }^{2}\theta =3\left({\sin }^{2}\theta +{\cos }^{2}\theta \right)+{\cos }^{2}\theta =3+{\cos }^{2}\theta$

Section 5.2 Solutions

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

3. $\sin \left(-x\right)=-\sin x$, so $\sin x$ is odd. $\cos \left(-x\right)=\cos \left(0-x\right)=\cos x$, so $\cos x$ is even.

5. $\frac{\sqrt{2}+\sqrt{6}}{4}$

7. $\frac{\sqrt{6}-\sqrt{2}}{4}$

9. $-2-\sqrt{3}$

11. $-\frac{\sqrt{2}}{2}\sin x-\frac{\sqrt{2}}{2}\cos x$

13. $-\frac{1}{2}\cos x-\frac{\sqrt{3}}{2}\sin x$

15. $\csc \theta$

17. $\cot x$

19. $\tan \left(\frac{x}{10}\right)$

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

23. $\frac{\sqrt{2}-\sqrt{6}}{4}$

25. $\sin x$

27. $\cot \left(\frac{\pi }{6}-x\right)$

29. $\cot \left(\frac{\pi }{4}+x\right)$

31. $\frac{\sin x}{\sqrt{2}}+\frac{\cos x}{\sqrt{2}}$

 

33. They are the same.

35. They are the different, try $g\left(x\right)=\sin \left(9x\right)-\cos \left(3x\right)\sin \left(6x\right)$.

37. They are the same.

39. They are the different, try $g\left(\theta \right)=\frac{2\tan \theta }{1-{\tan }^{2}\theta }$.

41. They are different, try $g\left(x\right)=\frac{\tan x-\tan \left(2x\right)}{1+\tan x\tan \left(2x\right)}$.

43. $-\frac{\sqrt{3}-1}{2\sqrt{2}},\text{ or }-0.2588$

45. $\frac{1+\sqrt{3}}{2\sqrt{2}}$, or 0.9659

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

 

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

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

53. True

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

Section 5.3 Solutions

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

3. $\frac{1-\cos x}{\sin x},\frac{\sin x}{1+\cos x}$, multiplying the top and bottom by $\sqrt{1-\cos x}$ and $\sqrt{1+\cos x}$, respectively.

5. a) $\frac{3\sqrt{7}}{32}$ b) $\frac{31}{32}$ c) $\frac{3\sqrt{7}}{31}$

7. a) $\frac{\sqrt{3}}{2}$ b) $-\frac{1}{2}$ c) $-\sqrt{3}$

9. $\cos \theta =-\frac{2\sqrt{5}}{5},\sin \theta =\frac{\sqrt{5}}{5},\tan \theta =-\frac{1}{2},\csc \theta =\sqrt{5},\sec \theta =-\frac{\sqrt{5}}{2},\cot \theta =-2$

11. $2\sin \left(\frac{\pi }{2}\right)=2$

13. $\frac{\sqrt{2-\sqrt{2}}}{2}$

15. $\frac{\sqrt{2-\sqrt{3}}}{2}$

17. $2+\sqrt{3}$

19. $-1-\sqrt{2}$

21. a) $\frac{3\sqrt{13}}{13}$ b) $-\frac{2\sqrt{13}}{13}$ c) $-\frac{3}{2}$

23. a) $\frac{\sqrt{10}}{4}$ b) $\frac{\sqrt{6}}{4}$ c) $\frac{\sqrt{15}}{3}$

25. $\frac{120}{169},-\frac{119}{169},-\frac{120}{119}$

27. $\frac{2\sqrt{13}}{13},\frac{3\sqrt{13}}{13},\frac{2}{3}$

29. $\cos \left({74}^{\circ }\right)$

31. $\cos \left(18x\right)$

33. $3\sin \left(10x\right)$

35. $-2\sin \left(-x\right)\cos \left(-x\right)=-2\left(-\sin \left(x\right)\cos \left(x\right)\right)=\sin \left(2x\right)$

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

39. $\frac{1+\cos \left(12x\right)}{2}$

41. $\frac{3+\cos \left(12x\right)-4\cos \left(6x\right)}{8}$

43. $\frac{2+\cos \left(2x\right)-2\cos \left(4x\right)-\cos \left(6x\right)}{32}$

45. $\frac{3+\cos \left(4x\right)-4\cos \left(2x\right)}{3+\cos \left(4x\right)+4\cos \left(2x\right)}$

47. $\frac{1-\cos \left(4x\right)}{8}$

49. $\frac{3+\cos \left(4x\right)-4\cos \left(2x\right)}{4\left(\cos \left(2x\right)+1\right)}$

51. $\frac{\left(1+\cos \left(4x\right)\right)\sin x}{2}$

53. $4\sin x\cos x\left({\cos }^{2}x-{\sin }^{2}x\right)$

55. $\frac{2\tan x}{1+{\tan }^{2}x}=\frac{\frac{2\sin x}{\cos x}}{1+\frac{{\sin }^{2}x}{{\cos }^{2}x}}=\frac{\frac{2\sin x}{\cos x}}{\frac{{\cos }^{2}x+{\sin }^{2}x}{{\cos }^{2}x}}=$
$\frac{2\sin x}{\cos x}.\frac{{\cos }^{2}x}{1}=2\sin x\cos x=\sin \left(2x\right)$

57. $\frac{2\sin x\cos x}{2{\cos }^{2}x - 1}=\frac{\sin \left(2x\right)}{\cos \left(2x\right)}=\tan \left(2x\right)$

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

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

63. $\begin{array}{l}\left({\cos }^{2}\left(4x\right)-{\sin }^{2}\left(4x\right)-\sin \left(8x\right)\right)\left({\cos }^{2}\left(4x\right)-{\sin }^{2}\left(4x\right)+\sin \left(8x\right)\right)=\hfill \\ \text{ }=\left(\cos \left(8x\right)-\sin \left(8x\right)\right)\left(\cos \left(8x\right)+\sin \left(8x\right)\right)\hfill \\ \text{ }={\cos }^{2}\left(8x\right)-{\sin }^{2}\left(8x\right)\hfill \\ \text{ }=\cos \left(16x\right)\hfill \\ \hfill \end{array}$

Section 5.4 Solutions

1. Substitute $\alpha$ into cosine and $\beta$ into sine and evaluate.

3. Answers will vary. There are some equations that involve a sum of two trig expressions where when converted to a product are easier to solve. For example: $\frac{\sin \left(3x\right)+\sin x}{\cos x}=1$. When converting the numerator to a product the equation becomes: $\frac{2\sin \left(2x\right)\cos x}{\cos x}=1\\$

5. $8\left(\cos \left(5x\right)-\cos \left(27x\right)\right)$

7. $\sin \left(2x\right)+\sin \left(8x\right)$

9. $\frac{1}{2}\left(\cos \left(6x\right)-\cos \left(4x\right)\right)$

11. $2\cos \left(5t\right)\cos t$

13. $2\cos \left(7x\right)$

15. $2\cos \left(6x\right)\cos \left(3x\right)$

17. $\frac{1}{4}\left(1+\sqrt{3}\right)$

19. $\frac{1}{4}\left(\sqrt{3}-2\right)$

21. $\frac{1}{4}\left(\sqrt{3}-1\right)$

23. $\cos \left(80^\circ \right)-\cos \left(120^\circ \right)$

25. $\frac{1}{2}\left(\sin \left(221^\circ \right)+\sin \left(205^\circ \right)\right)$

27. $\sqrt{2}\cos \left(31^\circ \right)$

29. $2\cos \left(66.5^\circ \right)\sin \left(34.5^\circ \right)$

31. $2\sin \left(-1.5^\circ \right)\cos \left(0.5^\circ \right)$

33. ${2}\sin \left({7x}\right){-2}\sin{ x}={ 2}\sin \left({4x}+{ 3x }\right)-{ 2 }\sin\left({4x } - { 3x }\right)=\\ {2}\left(\sin\left({ 4x }\right)\cos\left({ 3x }\right)+\sin\left({ 3x }\right)\cos\left({ 4x }\right)\right)-{ 2 }\left(\sin\left({ 4x }\right)\cos\left({ 3x }\right)-\sin \left({ 3x }\right)\cos\left({ 4x }\right)\right)=\\{2}\sin\left({ 4x }\right)\cos\left({ 3x }\right)+{2}\sin\left({ 3x }\right)\cos\left({ 4x }\right)-{ 2 }\sin\left({ 4x }\right)\cos\left({ 3x }\right)+{ 2 }\sin\left({ 3x }\right)\cos\left({ 4x }\right)=\\{ 4 }\sin\left({ 3x }\right)\cos\left({ 4x }\right)\\$

 

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

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

39. $2\cos \left({35}^{\circ }\right)\cos \left({23}^{\circ }\right),\text{ 1}\text{.5081}$

41. $-2\sin \left({33}^{\circ }\right)\sin \left({11}^{\circ }\right),\text{ }-0.2078$

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

45. It is an identity.

47. It is not an identity, but $2{\cos }^{3}x$ is.

49. $\tan \left(3t\right)$

51. $2\cos \left(2x\right)$

53. $-\sin \left(14x\right)$

55. Start with $\cos x+\cos y$. Make a substitution and let $x=\alpha +\beta$ and let $y=\alpha -\beta$, so $\cos x+\cos y$ becomes

$\cos \left(\alpha +\beta \right)+\cos \left(\alpha -\beta \right)=\cos \alpha \cos \beta -\sin \alpha \sin \beta +\cos \alpha \cos \beta +\sin \alpha \sin \beta =2\cos \alpha \cos \beta$

Since $x=\alpha +\beta$ and $y=\alpha -\beta$, we can solve for $\alpha$ and $\beta$ in terms of x and y and substitute in for $2\cos \alpha \cos \beta$ and get $2\cos \left(\frac{x+y}{2}\right)\cos \left(\frac{x-y}{2}\right)$.

57. $\frac{\cos \left(3x\right)+\cos x}{\cos \left(3x\right)-\cos x}=\frac{2\cos \left(2x\right)\cos x}{-2\sin \left(2x\right)\sin x}=-\cot \left(2x\right)\cot x$

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

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

63. $\tan \left(\frac{\pi }{4}-t\right)=\frac{\tan \left(\frac{\pi }{4}\right)-\tan t}{1+\tan \left(\frac{\pi }{4}\right)\tan \left(t\right)}=\frac{1-\tan t}{1+\tan t}$

Section 5.5 Solutions

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

3. If the sine or cosine function has a coefficient of one, isolate the term on one side of the equals sign. If the number it is set equal to has an absolute value less than or equal to one, the equation has solutions, otherwise it does not. If the sine or cosine does not have a coefficient equal to one, still isolate the term but then divide both sides of the equation by the leading coefficient. Then, if the number it is set equal to has an absolute value greater than one, the equation has no solution.

5. $\frac{\pi }{3},\frac{2\pi }{3}$

7. $\frac{3\pi }{4},\frac{5\pi }{4}$

9. $\frac{\pi }{4},\frac{5\pi }{4}$

11. $\frac{\pi }{4},\frac{3\pi }{4},\frac{5\pi }{4},\frac{7\pi }{4}$

13. $\frac{\pi }{4},\frac{7\pi }{4}$

15. $\frac{7\pi }{6},\frac{11\pi }{6}$

17. $\frac{\pi }{18},\frac{5\pi }{18},\frac{13\pi }{18},\frac{17\pi }{18},\frac{25\pi }{18},\frac{29\pi }{18}$

19. $\frac{3\pi }{12},\frac{5\pi }{12},\frac{11\pi }{12},\frac{13\pi }{12},\frac{19\pi }{12},\frac{21\pi }{12}$

21. $\frac{1}{6},\frac{5}{6},\frac{13}{6},\frac{17}{6},\frac{25}{6},\frac{29}{6},\frac{37}{6}$

23. $0,\frac{\pi }{3},\pi ,\frac{5\pi }{3}$

25. $\frac{\pi }{3},\pi ,\frac{5\pi }{3}$

27. $\frac{\pi }{3},\frac{3\pi }{2},\frac{5\pi }{3}$

29. $0,\pi$

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

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

35. $0$

37. $\frac{\pi }{6},\frac{5\pi }{6},\frac{7\pi }{6},\frac{11\pi }{6}$

39. $\frac{3\pi }{2},\frac{\pi }{6},\frac{5\pi }{6}$

41. $0,\frac{\pi }{3},\pi ,\frac{4\pi }{3}$

43. There are no solutions.

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

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

49. There are no solutions.

51. There are no solutions.

53. $0,\frac{2\pi }{3},\frac{4\pi }{3}$

55. $\frac{\pi }{4},\frac{3\pi }{4},\frac{5\pi }{4},\frac{7\pi }{4}$

57. ${\sin }^{-1}\left(\frac{3}{5}\right),\frac{\pi }{2},\pi -{\sin }^{-1}\left(\frac{3}{5}\right),\frac{3\pi }{2}$

59. ${\cos }^{-1}\left(-\frac{1}{4}\right),2\pi -{\cos }^{-1}\left(-\frac{1}{4}\right)$

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

63. ${\cos }^{-1}\left(\frac{3}{4}\right),{\cos }^{-1}\left(-\frac{2}{3}\right),2\pi -{\cos }^{-1}\left(-\frac{2}{3}\right),2\pi -{\cos }^{-1}\left(\frac{3}{4}\right)$

65. $0,\frac{\pi }{2},\pi ,\frac{3\pi }{2}$

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

69. There are no solutions.

71. $\pi +{\tan }^{-1}\left(-2\right),\pi +{\tan }^{-1}\left(-\frac{3}{2}\right),2\pi +{\tan }^{-1}\left(-2\right),2\pi +{\tan }^{-1}\left(-\frac{3}{2}\right)$

73. $2\pi k+0.2734,2\pi k+2.8682$

75. $\pi k - 0.3277$

77. $0.6694,1.8287,3.8110,4.9703$

79. $1.0472,3.1416,5.2360$

81. $0.5326,1.7648,3.6742,4.9064$

83. ${\sin }^{-1}\left(\frac{1}{4}\right),\pi -{\sin }^{-1}\left(\frac{1}{4}\right),\frac{3\pi }{2}$

85. $\frac{\pi }{2},\frac{3\pi }{2}$

87. There are no solutions.

89. $0,\frac{\pi }{2},\pi ,\frac{3\pi }{2}$

91. There are no solutions.

93. ${7.2}^{\circ }$

95. ${5.7}^{\circ }$

97. ${82.4}^{\circ }$

99. ${31.0}^{\circ }$

101. ${88.7}^{\circ }$

103. ${59.0}^{\circ }$

105. ${36.9}^{\circ }$

Section 5.6 Solutions

1. Physical behavior should be periodic, or cyclical.

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

5. $y=-3\cos \left(\frac{\pi }{6}x\right)-1$

7. $5\sin \left(2x\right)+2$

9. $4\cos \left(\frac{x\pi }{2}\right)-3$

11. $5 - 8\sin \left(\frac{x\pi }{2}\right)$

13. $\tan \left(\frac{x\pi }{12}\right)$

15. Answers will vary. Sample answer: This function could model temperature changes over the course of one very hot day in Phoenix, Arizona.

17. 9 years from now

19. $56^\circ \text{F}$

21. $1.8024$ 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: $\frac{1}{6}$, frequency: 6 Hz

33. Amplitude: 5, period: $\frac{1}{30}$, frequency: 30 Hz

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

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

39. $D\left(t\right)=7{\left(0.89\right)}^{t}\cos \left(40\pi t\right)$

41. $D\left(t\right)=19{\left(0.9265\right)}^{t}\cos \left(26\pi t\right)$

43. $20.1$ years

45. 17.8 seconds

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

49. 500 miles, at ${90}^{\circ }$

51. $y=6{\left(5\right)}^{x}+4\sin \left(\frac{\pi }{2}x\right)$

53. $y=8{\left(\frac{1}{2}\right)}^{x}\cos \left(\frac{\pi }{2}x\right)+3$