Advanced Mathematical Concepts Answers Chapter 7

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7.1 Solving Trigonometric Equations with Identities

1.

$$\begin{array}{l}\csc \theta \cos \theta \tan \theta =\left(\frac{1}{\sin \theta }\right)\cos \theta \left(\frac{\sin \theta }{\cos \theta }\right)\hfill \\ =\frac{\cos \theta }{\sin \theta }\left(\frac{\sin \theta }{\cos \theta }\right)\hfill \\ =\frac{\sin \theta \cos \theta }{\sin \theta \cos \theta }\hfill \\ =1\hfill \end{array}$$

2.

$$\begin{array}{l}\frac{\cot \theta }{\csc \theta }=\frac{\frac{\cos \theta }{\sin \theta }}{\frac{1}{\sin \theta }}\hfill \\ =\frac{\cos \theta }{\sin \theta }\cdot \frac{\sin \theta }{1}\hfill \\ =\cos \theta \hfill \end{array}$$

3.

$\begin{array}{c}\frac{{\sin }^2\theta -1}{\tan \theta \sin \theta -\tan \theta }=\frac{\left(\sin \theta +1\right)\left(\sin \theta -1\right)}{\tan \theta \left(\sin \theta -1\right)}\\ =\frac{\sin \theta +1}{\tan \theta }\end{array}$

4.

This is a difference of squares formula: $25-9{\sin }^2\theta =(5-3\sin \theta )(5+3\sin \theta ).$

5.

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

7.2 Sum and Difference Identities

4.

$\cos \left(\frac{5\pi }{14}\right)$

5.

$$\begin{array}{l}\tan (\pi -\theta )=\frac{\tan (\pi )-\tan \theta }{1+\tan (\pi )\tan \theta }\hfill \\ =\frac{0-\tan \theta }{1+0\cdot \tan \theta }\hfill \\ =-\tan \theta \hfill \end{array}$$

7.3 Double-Angle, Half-Angle, and Reduction Formulas

1.

$\cos \left(2\alpha \right)=\frac{7}{32}$

2.

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

3.

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

4.

$$\begin{array}{ll}10{\cos }^{4}x=10{\cos }^{4}x=10{({\cos }^{2}x)}^2\hfill & \hfill \\ =10{\left[\frac{1+\cos (2x)}{2}\right]}^2\hfill & {\text{Substitute reduction formula for cos}}^{2}x.\hfill \\ =\frac{10}{4}[1+2\cos (2x)+{\cos }^2(2x)]\hfill & \hfill \\ =\frac{10}{4}+\frac{10}{2}\cos (2x)+\frac{10}{4}\left(\frac{1+\cos 2(2x)}{2}\right)\hfill & {\text{Substitute reduction formula for cos}}^{2}x.\hfill \\ =\frac{10}{4}+\frac{10}{2}\cos (2x)+\frac{10}{8}+\frac{10}{8}\cos (4x)\hfill & \hfill \\ =\frac{30}{8}+5\cos (2x)+\frac{10}{8}\cos (4x)\hfill & \hfill \\ =\frac{15}{4}+5\cos (2x)+\frac{5}{4}\cos (4x)\hfill & \hfill \end{array}$$

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

1.

$\frac{1}{2}\left(\cos 6\theta +\cos 2\theta \right)$

2.

$\frac{1}{2}\left(\sin 2x+\sin 2y\right)$

4.

$2\sin \left(2\theta \right)\cos \left(\theta \right)$

5.

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

7.5 Solving Trigonometric Equations

1.

$x=\frac{7\pi }{6},\frac{11\pi }{6}$

3.

$\theta \approx 1.7722\pm 2\pi k$ and $\theta \approx 4.5110\pm 2\pi k$

4.

$\cos \theta =-1,\theta =\pi$

5.

$\frac{\pi }{2},\frac{2\pi }{3},\frac{4\pi }{3},\frac{3\pi }{2}$

7.6 Modeling with Trigonometric Functions

1.

The amplitude is $3,$ and the period is $\frac{2}{3}.$

2.

x	$3\sin \left(3x\right)$
0	0
$\frac{\pi }{6}$	3
$\frac{\pi }{3}$	0
$\frac{\pi }{2}$	$\mathrm{-3}$
$\frac{2\pi }{3}$	0

3.

$y=8\sin \left(\frac{\pi }{12}t\right)+32$
The temperature reaches freezing at noon and at midnight.

4.

initial displacement =6, damping constant = -6, frequency = $\frac{2}{\pi }$

5.

$y=10e^{-0.5t}\cos \left(\pi t\right)$

6.

$y=5\cos \left(6\pi t\right)$

7.1 Section Exercises

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.

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

39.

Proved with negative and Pythagorean Identities

41.

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

7.2 Section Exercises

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.

11.

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

13.

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

19.

$\tan \left(\frac{x}{10}\right)$

21.

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

25.

$\sin x$

27.

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

29.

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

31.

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

35.

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

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.

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

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.

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

55.

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

7.3 Section Exercises

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.

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

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.

$\cos ({74}^{\circ })$

35.

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

37.

$\begin{array}{l}\frac{\sin \left(2\theta \right)}{1+\cos \left(2\theta \right)}{\tan }^2\theta =\frac{2\sin \left(\theta \right)\cos \left(\theta \right)}{1+{\cos }^2\theta -{\sin }^2\theta }{\tan }^2\theta =\\ \frac{2\sin \left(\theta \right)\cos \left(\theta \right)}{2{\cos }^2\theta }{\tan }^2\theta =\frac{\sin \left(\theta \right)}{\cos \theta }{\tan }^2\theta =\\ \tan \theta {\tan }^2\theta ={\tan }^3\theta \end{array}$

39.

$\frac{1+\cos (12x)}{2}$

41.

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

43.

$\frac{2+\cos (2x)-2\cos (4x)-\cos (6x)}{32}$

45.

$\frac{3+\cos (4x)-4\cos (2x)}{3+\cos (4x)+4\cos (2x)}$

47.

$\frac{1-\cos (4x)}{8}$

49.

$\frac{3+\cos (4x)-4\cos (2x)}{4(\cos (2x)+1)}$

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.

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

57.

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

59.

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

61.

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

63.

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

7.4 Section Exercises

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: $\frac{\sin (3x)+\sin x}{\cos x}=1.$ When converting the numerator to a product the equation becomes: $\frac{2\sin (2x)\cos x}{\cos x}=1$

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.

$\cos \left(\mathrm{80°}\right)-\cos \left(\mathrm{120°}\right)$

25.

$\frac{1}{2}(\sin (\mathrm{221°})+\sin (\mathrm{205°}))$

27.

$\sqrt{2}\cos \left(\mathrm{31°}\right)$

29.

$2\cos (66.5°)\sin (34.5°)$

31.

$2\sin \left(\mathrm{-1.5°}\right)\cos \left(\mathrm{0.5°}\right)$

33.

$$\begin{array}{l}2\sin (7x)-2\sin x=2\sin (4x+3x)-2\sin (4x-3x)=\hfill \\ 2(\sin (4x)\cos (3x)+\sin (3x)\cos (4x))-2(\sin (4x)\cos (3x)-\sin (3x)\cos (4x))=\hfill \\ 2\sin (4x)\cos (3x)+2\sin (3x)\cos (4x))-2\sin (4x)\cos (3x)+2\sin (3x)\cos (4x))=\hfill \\ 4\sin (3x)\cos (4x)\hfill \\ \hfill \end{array}$$

35.

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

37.

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

39.

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

41.

$-2\sin ({33}^{\circ })\sin ({11}^{\circ }),-0.2078$

43.

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

47.

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

51.

$2\cos \left(2x\right)$

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

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.

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

63.

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

7.5 Section Exercises

1.

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

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.

7.

$\frac{3\pi }{4},\frac{5\pi }{4}$

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.

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

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.

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

37.

$\theta ={\sin }^{-1}\left(\frac{2}{3}\right),\mathrm{\pi }-{\sin }^{-1}\left(\frac{2}{3}\right),\mathrm{\pi }+{\sin }^{-1}\left(\frac{2}{3}\right),2\mathrm{\pi }-{\sin }^{-1}\left(\frac{2}{3}\right)$

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.

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

69.

There are no solutions.

71.

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

73.

$2\pi k+0.2734,2\pi k+2.8682$

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.

7.6 Section Exercises

1.

Physical behavior should be periodic, or cyclical.

3.

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

5.

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

8.

$y=4-6\cos \left(\frac{x\pi }{2}\right)$

10.

$y=\tan \left(\frac{x\pi }{8}\right)$

12.

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

23.

From June 15 through November 16

25.

From day 31 through day 58

27.

Floods: April 16 to July 15. Drought: October 16 to January 15.

29.

Amplitude: 8, period: $\frac{1}{3},$ frequency: 3 Hz

31.

Amplitude: 4, period: $4,$ frequency: $\frac{1}{4}$ Hz

33.

$P(t)=-19\cos \left(\frac{\pi }{6}t\right)+800+\frac{160}{12}tP(t)=-19\cos \left(\frac{\pi }{6}t\right)+800+\frac{40}{3}t$

35.

$P(t)=-33\cos \left(\frac{\pi }{6}t\right)+900+{\left(1.07\right)}^{}t$

37.

$D(t)=10{\left(0.85\right)}^t\cos \left(36\pi t\right)$

39.

$D(t)=17{\left(0.9145\right)}^t\cos \left(28\pi t\right)$

45.

Spring 2 comes to rest first after 7.3 seconds.

47.

234.3 miles, at 72.2°

49.

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

51.

$y=4{(–2)}^x+8\sin \left(\frac{\pi }{2}x\right)$

53.

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

Review Exercises

1.

${\sin }^{-1}\left(\frac{\sqrt{3}}{3}\right),\pi -{\sin }^{-1}\left(\frac{\sqrt{3}}{3}\right),\pi +{\sin }^{-1}\left(\frac{\sqrt{3}}{3}\right),2\pi -{\sin }^{-1}\left(\frac{\sqrt{3}}{3}\right)$

3.

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

5.

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

15.

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

17.

$\tan \left(\frac{5}{8}x\right)$

21.

$-\frac{24}{25},-\frac{7}{25},\frac{24}{7}$

25.

$\frac{\sqrt{2}}{10},\frac{7\sqrt{2}}{10},\frac{1}{7},\frac{3}{5},\frac{4}{5},\frac{3}{4}$

27.

$\begin{array}{l}\hfill \\ \cot x\cos (2x)=\cot x(1-2{\sin }^{2}x)\hfill \\ =\cot x-\frac{\cos x}{\sin x}(2){\sin }^{2}x\hfill \\ =-2\sin x\cos x+\cot x\hfill \\ =-\sin (2x)+\cot x\hfill \end{array}$

29.

$\frac{10\sin x-5\sin \left(3x\right)+\sin \left(5x\right)}{8\left(\cos \left(2x\right)+1\right)}$

35.

$\frac{1}{2}\left(\sin (6x)+\sin (12x)\right)$

37.

$2\sin \left(\frac{13}{2}x\right)\cos \left(\frac{9}{2}x\right)$

39.

$\frac{3\pi }{4},\frac{7\pi }{4}$

41.

$0,\frac{\pi }{6},\frac{5\pi }{6},\pi$

47.

$0.2527,2.8889,4.7124$

49.

$1.3694$ , $1.9106$ , $4.3726$ , $4.9137$

51.

$3\sin \left(\frac{x\pi }{2}\right)-2$

55.

$P(t)=950-450\sin \left(\frac{\pi }{6}t\right)$

57.

Amplitude: 3, period: 2, frequency: $\frac{1}{2}$ Hz

59.

$C(t)=20\sin (2\pi t)+100{(1.4427)}^t$

Practice Test

11.

$2\text{cos}\left(3x\right)\text{cos}\left(5x\right)$

13.

$x={\text{cos}}^{\mathrm{–1}}\left(\frac{1}{5}\right)$

15.

$\frac{3}{5},-\frac{4}{5},-\frac{3}{4}$

17.

$\begin{array}{cc}\hfill {\text{tan}}^{3}x–\text{tan}x{\text{sec}}^{2}x& =\text{tan}x({\text{tan}}^{2}x–{\text{sec}}^{2}x)\hfill \\ \hfill & =\text{tan}x({\text{tan}}^{2}x–(1+{\text{tan}}^{2}x\left)\right)\hfill \\ \hfill & =\text{tan}x({\text{tan}}^{2}x–1–{\text{tan}}^{2}x)\hfill \\ \hfill & =–\text{tan}x=\text{tan}(–x)=\text{tan}–x)\hfill \end{array}$

19.

$\begin{array}{cc}\hfill \frac{\text{sin}\left(2x\right)}{\text{sin}x}–\frac{\text{cos}\left(2x\right)}{\text{cos}x}& =\frac{2\text{sin}x\text{cos}x}{\text{sin}x}–\frac{2{\text{cos}}^{2}x–1}{\text{cos}x}\hfill \\ \hfill & =2\text{cos}x–2\text{cos}x+\frac{1}{\text{cos}x}\hfill \\ \hfill & =\frac{1}{\text{cos}x}=\text{sec}x=\text{sec}x\hfill \end{array}$

21.

Amplitude: $\frac{1}{4}$ , period $\frac{1}{60}$ , frequency: 60 Hz

23.

Amplitude: $8$ , fast period: $\frac{1}{500}$ , fast frequency: 500 Hz, slow period: $\frac{1}{10}$ , slow frequency: 10 Hz

25.

$D\left(t\right)=20\left(0.9086{)}^t\text{cos}\right(4\pi t)$ , 31 seconds

Advanced Mathematical Concepts Answers Chapter 7

Source: https://openstax.org/books/precalculus/pages/chapter-7

Posted by: lynnbereest.blogspot.com

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