Chapter 7 Test Answer Key

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7.i Solving Trigonometric Equations with Identities

1.

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

2.

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

3.

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

iv.

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

5.

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

seven.2 Sum and Difference Identities

4.

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

5.

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

7.iii Double-Angle, Half-Angle, and Reduction Formulas

1.

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

2.

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

3.

$\cos \left(\mathrm{two}\theta \right)\cos \theta =\left({\cos }^{\mathrm{two}}\theta -{\sin }^2\theta \right)\cos \theta ={\cos }^{\mathrm{iii}}\theta -\cos \theta {\sin }^2\theta$

4.

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

7.four Sum-to-Product and Production-to-Sum Formulas

1.

$\frac{\mathrm{one}}{2}\left(\cos \mathrm{six}\theta +\cos 2\theta \right)$

2.

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

4.

$\mathrm{ii}\sin \left(\mathrm{ii}\theta \right)\cos \left(\theta \right)$

5.

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

7.5 Solving Trigonometric Equations

1.

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

3.

$\theta \approx \mathrm{i.7722}\pm 2\pi k$ and $\theta \approx 4.5110\pm 2\pi \mathrm{grand}$

4.

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

5.

$\frac{\pi }{\mathrm{two}},\frac{2\pi }{3},\frac{4\pi }{3},\frac{\mathrm{three}\pi }{2}$

7.half dozen Modeling with Trigonometric Functions

1.

The amplitude is $\mathrm{iii},$ and the flow is $\frac{\mathrm{ii}}{3}.$

2.

x	$3\sin \left(3x\right)$
0	0
$\frac{\pi }{\mathrm{half-dozen}}$	3
$\frac{\pi }{\mathrm{three}}$	0
$\frac{\pi }{2}$	$\mathrm{-three}$
$\frac{\mathrm{ii}\pi }{\mathrm{three}}$	0

three.

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

iv.

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

5.

$y=\mathrm{x}{e}^{-\mathrm{0.v}t}\cos \left(\pi t\right)$

6.

$y=\mathrm{v}\cos \left(\mathrm{half-dozen}\pi t\right)$

vii.1 Section Exercises

1.

All three functions, $F$ , $K$ , and $H,$ are even.

This is because $F\left(-x\right)=\sin \left(-x\right)\sin \left(-x\right)=\left(-\sin \mathrm{ten}\right)\left(-\sin \mathrm{10}\right)={\sin }^{2}x=F\left(x\right),\mathrm{Grand}\left(-\mathrm{10}\right)=\cos \left(-\mathrm{ten}\right)\cos \left(-x\right)=\cos \mathrm{10}\cos \mathrm{ten}={\cos }^{2}x=M\left(\mathrm{10}\right)$ and $H\left(-\mathrm{10}\right)=\tan \left(-\mathrm{ten}\right)\tan \left(-\mathrm{ten}\right)=\left(-\tan \mathrm{ten}\right)\left(-\tan x\right)={\tan }^{\mathrm{two}}\mathrm{ten}=H\left(x\right).$

iii.

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

23.

$-4\sec x\tan \mathrm{ten}$

25.

$\pm \sqrt{\frac{\mathrm{i}}{{\cot }^2\mathrm{ten}}+1}$

27.

$\frac{\pm \sqrt{\mathrm{one}-{\sin }^{2}x}}{\sin x}$

29.

Answers will vary. Sample proof:

$\cos x-{\cos }^{\mathrm{iii}}x=\cos x\left(1-{\cos }^{2}x\right)$
$=\cos x{\sin }^2\mathrm{ten}$

31.

Answers will vary. Sample proof:
$\frac{1+{\sin }^2\mathrm{ten}}{{\cos }^{2}x}=\frac{1}{{\cos }^{2}x}+\frac{{\sin }^{2}x}{{\cos }^{\mathrm{ii}}\mathrm{ten}}={\sec }^{2}x+{\tan }^2\mathrm{ten}={\tan }^{2}x+1+{\tan }^2\mathrm{10}=\mathrm{i}+2{\tan }^{2}x$

33.

Answers will vary. Sample proof:
${\cos }^{\mathrm{two}}x-{\tan }^2\mathrm{10}=1-{\sin }^2\mathrm{ten}-\left({\sec }^{2}x-\mathrm{ane}\right)=\mathrm{ane}-{\sin }^{\mathrm{two}}\mathrm{10}-{\sec }^{2}x+\mathrm{i}=2-{\sin }^{2}x-{\sec }^{\mathrm{two}}x$

39.

Proved with negative and Pythagorean Identities

41.

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

7.2 Section Exercises

one.

The cofunction identities utilize to complementary angles. Viewing the 2 acute angles of a correct triangle, if one of those angles measures $x,$ the second bending measures $\frac{\pi }{\mathrm{ii}}-x.$ And then $\sin x=\cos \left(\frac{\pi }{2}-\mathrm{10}\right).$ The same holds for the other cofunction identities. The fundamental is that the angles are complementary.

3.

$\sin \left(-x\right)=-\sin x,$ so $\sin x$ is odd. $\cos \left(-x\right)=\cos \left(0-x\right)=\cos \mathrm{10},$ and then $\cos x$ is even.

xi.

$-\frac{\sqrt{\mathrm{ii}}}{\mathrm{ii}}\sin \mathrm{10}-\frac{\sqrt{2}}{2}\cos x$

thirteen.

$-\frac{1}{2}\cos \mathrm{10}-\frac{\sqrt{3}}{2}\sin x$

19.

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

21.

$\sin (a-b)=\left(\frac{4}{5}\right)\left(\frac{1}{\mathrm{three}}\right)-\left(\frac{3}{\mathrm{v}}\right)\left(\frac{2\sqrt{2}}{3}\right)=\frac{\mathrm{four}-\mathrm{vi}\sqrt{2}}{\mathrm{fifteen}}$
$\cos (a+b)=\left(\frac{3}{5}\right)\left(\frac{\mathrm{ane}}{\mathrm{three}}\right)-\left(\frac{\mathrm{four}}{5}\right)\left(\frac{2\sqrt{2}}{3}\right)=\frac{3-8\sqrt{2}}{\mathrm{xv}}$

25.

$\sin x$

27.

$\cot \left(\frac{\pi }{6}-\mathrm{10}\right)$

29.

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

31.

$\frac{\sin x}{\sqrt{\mathrm{ii}}}+\frac{\cos x}{\sqrt{\mathrm{ii}}}$

35.

They are the different, endeavour $\mathrm{1000}\left(x\right)=\sin \left(\mathrm{ix}x\right)-\cos \left(\mathrm{iii}\mathrm{10}\right)\sin \left(\mathrm{vi}x\right).$

39.

They are the different, try $m\left(\theta \right)=\frac{2\tan \theta }{\mathrm{one}-{\tan }^{\mathrm{ii}}\theta }.$

41.

They are unlike, effort $g\left(\mathrm{10}\right)=\frac{\tan x-\tan \left(\mathrm{two}x\right)}{\mathrm{one}+\tan \mathrm{10}\tan \left(2x\right)}.$

43.

$-\frac{\sqrt{\mathrm{iii}}-1}{2\sqrt{2}},\text{ or}-0.2588$

45.

$\frac{\mathrm{one}+\sqrt{\mathrm{three}}}{\mathrm{two}\sqrt{2}},$ or 0.9659

47.

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

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.

$\begin{array}{c}\frac{\cos \left(\mathrm{ten}+h\right)-\cos x}{h}=\\ \frac{\cos \mathrm{10}\cosh -\sin x\sinh -\cos x}{h}=\\ \frac{\cos x(\cosh -\mathrm{one})-\sin x\sinh }{h}=\cos x\frac{\cos h-1}{h}-\sin x\frac{\sin h}{h}\end{array}$

55.

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

7.iii Department Exercises

i.

Utilize the Pythagorean identities and isolate the squared term.

3.

$\frac{1-\cos x}{\sin \mathrm{ten}},\frac{\sin x}{\mathrm{one}+\cos x},$ multiplying the top and lesser by $\sqrt{1-\cos x}$ and $\sqrt{\mathrm{i}+\cos \mathrm{10}},$ respectively.

5.

a) $\frac{\mathrm{iii}\sqrt{7}}{32}$ b) $\frac{31}{32}$ c) $\frac{3\sqrt{7}}{31}$

7.

a) $\frac{\sqrt{\mathrm{three}}}{2}$ b) $-\frac{\mathrm{i}}{2}$ c) $-\sqrt{3}$

ix.

$\cos \theta =-\frac{\mathrm{two}\sqrt{\mathrm{v}}}{5},\sin \theta =\frac{\sqrt{5}}{5},\tan \theta =-\frac{\mathrm{one}}{\mathrm{two}},\csc \theta =\sqrt{5},\sec \theta =-\frac{\sqrt{\mathrm{v}}}{2},\cot \theta =-\mathrm{two}$

xi.

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

21.

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

23.

a) $\frac{\sqrt{\mathrm{x}}}{4}$ b) $\frac{\sqrt{\mathrm{six}}}{4}$ c) $\frac{\sqrt{\mathrm{xv}}}{\mathrm{iii}}$

25.

$\frac{120}{169},–\frac{119}{169},–\frac{120}{119}$

27.

$\frac{\mathrm{ii}\sqrt{13}}{\mathrm{thirteen}},\frac{3\sqrt{13}}{13},\frac{\mathrm{two}}{3}$

29.

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

35.

$-2\sin \left(-x\right)\cos \left(-\mathrm{ten}\right)=-\mathrm{ii}(-\sin \left(\mathrm{10}\right)\cos \left(x\right))=\sin \left(2x\right)$

37.

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

39.

$\frac{\mathrm{one}+\cos (12\mathrm{10})}{2}$

41.

$\frac{\mathrm{iii}+\cos (12\mathrm{10})-\mathrm{four}\cos (6\mathrm{ten})}{\mathrm{eight}}$

43.

$\frac{\mathrm{two}+\cos (\mathrm{two}x)-\mathrm{ii}\cos (4\mathrm{10})-\cos (\mathrm{half\; dozen}\mathrm{10})}{32}$

45.

$\frac{3+\cos (4x)-4\cos (2\mathrm{10})}{3+\cos (4\mathrm{10})+4\cos (2x)}$

47.

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

49.

$\frac{\mathrm{iii}+\cos (4x)-4\cos (2\mathrm{ten})}{4(\cos (\mathrm{two}\mathrm{10})+1)}$

51.

$\frac{\left(1+\cos \left(\mathrm{four}\mathrm{ten}\right)\right)\sin \mathrm{ten}}{2}$

53.

$4\sin x\cos x\left({\cos }^2\mathrm{ten}-{\sin }^{\mathrm{two}}x\right)$

55.

$\frac{2\tan x}{\mathrm{ane}+{\tan }^2\mathrm{ten}}=\frac{\frac{\mathrm{ii}\sin x}{\cos x}}{1+\frac{{\sin }^2\mathrm{ten}}{{\cos }^{\mathrm{two}}x}}=\frac{\frac{\mathrm{ii}\sin \mathrm{10}}{\cos x}}{\frac{{\cos }^{2}x+{\sin }^{\mathrm{ii}}x}{{\cos }^{\mathrm{ii}}x}}=$
$\frac{\mathrm{ii}\sin \mathrm{10}}{\cos x}.\frac{{\cos }^{2}x}{\mathrm{i}}=2\sin x\cos x=\sin (2x)$

57.

$\frac{2\sin x\cos x}{\mathrm{ii}{\cos }^{2}x-1}=\frac{\sin (2\mathrm{10})}{\cos (2\mathrm{10})}=\tan (2\mathrm{ten})$

59.

$\begin{array}{l}\sin (x+2\mathrm{10})=\sin x\cos (2\mathrm{10})+\sin (2x)\cos \mathrm{ten}\hfill \\ =\sin \mathrm{ten}({\cos }^2\mathrm{10}-{\sin }^{\mathrm{two}}x)+2\sin x\cos \mathrm{10}\cos x\hfill \\ =\sin x{\cos }^2\mathrm{ten}-{\sin }^{\mathrm{iii}}x+\mathrm{ii}\sin x{\cos }^{\mathrm{two}}x\hfill \\ =\mathrm{three}\sin x{\cos }^{2}x-{\sin }^{3}x\hfill \end{array}$

61.

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

63.

$\begin{array}{l}\left({\cos }^{\mathrm{ii}}(\mathrm{iv}x)-{\sin }^2(4x)-\sin (\mathrm{eight}x))({\cos }^{\mathrm{ii}}(4\mathrm{10})-{\sin }^{\mathrm{two}}(4x)+\sin (8\mathrm{10})\right)=\hfill \\ =\left(\cos (8x)-\sin (8x))(\cos (8x)+\sin (8\mathrm{10})\right)\hfill \\ ={\cos }^{\mathrm{two}}(8x)-{\sin }^{\mathrm{two}}(\mathrm{viii}x)\hfill \\ =\cos (\mathrm{sixteen}x)\hfill \end{array}$

7.4 Section Exercises

1.

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

3.

Answers volition vary. In that location are some equations that involve a sum of two trig expressions where when converted to a product are easier to solve. For example: $\frac{\sin (\mathrm{iii}\mathrm{10})+\sin \mathrm{ten}}{\cos x}=\mathrm{ane.}$ When converting the numerator to a product the equation becomes: $\frac{2\sin (2x)\cos x}{\cos \mathrm{ten}}=1$

5.

$\mathrm{viii}\left(\cos \left(5x\right)-\cos \left(27x\right)\right)$

vii.

$\sin \left(2\mathrm{ten}\right)+\sin \left(8x\right)$

9.

$\frac{1}{\mathrm{two}}\left(\cos \left(\mathrm{vi}x\right)-\cos \left(4\mathrm{10}\right)\right)$

11.

$\mathrm{two}\cos \left(5t\right)\cos t$

13.

$2\cos \left(\mathrm{seven}x\right)$

15.

$\mathrm{ii}\cos \left(6\mathrm{ten}\right)\cos \left(3x\right)$

17.

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

19.

$\frac{1}{4}\left(\sqrt{\mathrm{three}}-2\right)$

21.

$\frac{\mathrm{ane}}{4}\left(\sqrt{\mathrm{three}}-1\right)$

23.

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

25.

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

27.

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

29.

$\mathrm{ii}\cos (66.5°)\sin (\mathrm{34.five}°)$

31.

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

33.

$$\begin{array}{l}2\sin (7\mathrm{ten})-2\sin x=2\sin (4\mathrm{ten}+3x)-2\sin (4x-3x)=\hfill \\ 2(\sin (\mathrm{iv}\mathrm{10})\cos (\mathrm{three}x)+\sin (3\mathrm{10})\cos (4\mathrm{10}))-2(\sin (\mathrm{iv}x)\cos (3x)-\sin (3x)\cos (\mathrm{iv}x))=\hfill \\ 2\sin (4x)\cos (3x)+\mathrm{two}\sin (\mathrm{iii}x)\cos (\mathrm{four}\mathrm{10}))-2\sin (4x)\cos (\mathrm{three}\mathrm{10})+2\sin (\mathrm{three}x)\cos (4\mathrm{ten}))=\hfill \\ \mathrm{iv}\sin (3x)\cos (4x)\hfill \\ \hfill \end{array}$$

35.

$\sin x+\sin \left(3x\right)=2\sin \left(\frac{4x}{\mathrm{ii}}\right)\cos \left(\frac{-2x}{2}\right)=$
$\mathrm{two}\sin (\mathrm{ii}\mathrm{ten})\cos \mathrm{ten}=2(2\sin x\cos x)\cos \mathrm{ten}=$
$4\sin \mathrm{ten}{\cos }^2\mathrm{10}$

37.

$2\tan \mathrm{ten}\cos \left(3x\right)=\frac{2\sin x\cos (3\mathrm{ten})}{\cos \mathrm{ten}}=\frac{2(.5(\sin (\mathrm{iv}\mathrm{ten})-\sin (\mathrm{two}\mathrm{ten})))}{\cos x}$
$=\frac{\mathrm{ane}}{\cos x}\left(\sin (\mathrm{iv}\mathrm{ten})-\sin (2\mathrm{ten})\right)=\sec \mathrm{10}\left(\sin \left(4x\right)-\sin \left(2x\right)\right)$

39.

$\mathrm{ii}\cos ({35}^{\circ })\cos ({23}^{\circ }),\text{ i}\text{.5081}$

41.

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

43.

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

47.

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

51.

$\mathrm{ii}\cos \left(2\mathrm{10}\right)$

55.

Start with $\cos \mathrm{10}+\cos y.$ Make a commutation and let $x=\alpha +\beta$ and let $y=\alpha -\beta ,$ so $\cos x+\cos y$ becomes
$\cos (\alpha +\beta )+\cos (\alpha -\beta )=\cos \alpha \cos \beta -\sin \alpha \sin \beta +\cos \alpha \cos \beta +\sin \alpha \sin \beta =\mathrm{two}\cos \alpha \cos \beta$

Since $x=\alpha +\beta$ and $y=\alpha -\beta ,$ we can solve for $\alpha$ and $\beta$ in terms of ten and y and substitute in for $2\cos \alpha \cos \beta$ and get $2\cos \left(\frac{\mathrm{10}+y}{2}\right)\cos \left(\frac{x-y}{2}\right).$

57.

$\frac{\cos \left(3\mathrm{10}\right)+\cos x}{\cos \left(3x\right)-\cos \mathrm{10}}=\frac{\mathrm{two}\cos \left(\mathrm{two}x\right)\cos x}{-2\sin \left(2x\right)\sin x}=-\cot \left(2x\right)\cot x$

59.

$\begin{array}{l}\frac{\cos \left(2y\right)-\cos \left(4y\right)}{\sin \left(\mathrm{ii}y\right)+\sin \left(4y\right)}=\frac{-\mathrm{two}\sin \left(3y\right)\sin \left(-y\right)}{2\sin \left(\mathrm{iii}y\right)\cos y}=\\ \frac{\mathrm{two}\sin \left(\mathrm{three}y\right)\sin \left(y\right)}{2\sin \left(3y\right)\cos y}=\tan y\end{array}$

61.

$\begin{array}{l}\cos \mathrm{10}-\cos \left(3x\right)=-2\sin (\mathrm{two}\mathrm{10})\sin (-x)=\\ 2(2\sin x\cos x)\sin \mathrm{ten}=\mathrm{iv}{\sin }^{\mathrm{ii}}x\cos x\end{array}$

63.

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

7.v Section Exercises

i.

In that location will not ever be solutions to trigonometric function equations. For a bones example, $\cos (x)=\mathrm{-v.}$

3.

If the sine or cosine part has a coefficient of one, isolate the term on one side of the equals sign. If the number information technology is set equal to has an absolute value less than or equal to i, the equation has solutions, otherwise it does not. If the sine or cosine does not have a coefficient equal to 1, still isolate the term just then dissever 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.

seven.

$\frac{3\pi }{4},\frac{\mathrm{five}\pi }{\mathrm{iv}}$

eleven.

$\frac{\pi }{4},\frac{3\pi }{4},\frac{5\pi }{\mathrm{four}},\frac{7\pi }{4}$

thirteen.

$\frac{\pi }{4},\frac{\mathrm{seven}\pi }{4}$

15.

$\frac{\mathrm{seven}\pi }{6},\frac{11\pi }{6}$

17.

$\frac{\pi }{\mathrm{xviii}}$ , $\frac{\mathrm{v}\pi }{\mathrm{eighteen}}$ , $\frac{13\pi }{\mathrm{xviii}}$ , $\frac{17\pi }{18}$ , $\frac{25\pi }{\mathrm{xviii}}$ , $\frac{29\pi }{\mathrm{eighteen}}$

19.

$\frac{3\pi }{12},\frac{5\pi }{12},\frac{\mathrm{xi}\pi }{12},\frac{\mathrm{xiii}\pi }{12},\frac{\mathrm{nineteen}\pi }{12},\frac{21\pi }{12}$

21.

$\frac{1}{6},\frac{\mathrm{five}}{\mathrm{half-dozen}},\frac{\mathrm{xiii}}{6},\frac{17}{6},\frac{25}{\mathrm{six}},\frac{29}{6},\frac{37}{\mathrm{six}}$

23.

$0,\frac{\pi }{\mathrm{iii}},\pi ,\frac{\mathrm{v}\pi }{3}$

25.

$\frac{\pi }{\mathrm{three}},\pi ,\frac{5\pi }{3}$

27.

$\frac{\pi }{3},\frac{\mathrm{iii}\pi }{2},\frac{\mathrm{v}\pi }{3}$

31.

$\pi -{\sin }^{-1}\left(-\frac{1}{4}\right),\frac{7\pi }{6},\frac{11\pi }{6},\mathrm{ii}\pi +{\sin }^{-\mathrm{ane}}\left(-\frac{\mathrm{i}}{4}\right)$

33.

$$\frac{1}{3}\left({\sin }^{-\mathrm{ane}}\left(\frac{9}{10}\right)\right),\frac{\pi }{\mathrm{iii}}-\frac{\mathrm{one}}{3}\left({\sin }^{-\mathrm{one}}\left(\frac{9}{10}\right)\right),\frac{\mathrm{two}\pi }{3}+\frac{\mathrm{one}}{\mathrm{iii}}\left({\sin }^{-1}\left(\frac{9}{\mathrm{ten}}\right)\right),\pi -\frac{1}{3}\left({\sin }^{-\mathrm{ane}}\left(\frac{9}{10}\right)\right),\frac{\mathrm{four}\pi }{3}+\frac{1}{\mathrm{three}}\left({\sin }^{-1}\left(\frac{9}{10}\right)\right),\frac{\mathrm{five}\pi }{3}-\frac{\mathrm{ane}}{\mathrm{three}}\left({\sin }^{-1}\left(\frac{9}{10}\right)\right)$$

37.

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

39.

$\frac{\mathrm{three}\pi }{2},\frac{\pi }{\mathrm{vi}},\frac{5\pi }{6}$

41.

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

43.

There are no solutions.

45.

${\cos }^{-\mathrm{i}}\left(\frac{1}{3}\left(1-\sqrt{7}\right)\right),2\pi -{\cos }^{-1}\left(\frac{1}{\mathrm{iii}}\left(\mathrm{ane}-\sqrt{\mathrm{vii}}\right)\right)$

47.

${\tan }^{-1}\left(\frac{\mathrm{ane}}{2}\left(\sqrt{29}-5\right)\right),\pi +{\tan }^{-1}\left(\frac{1}{\mathrm{two}}\left(-\sqrt{29}-5\right)\right),\pi +{\tan }^{-1}\left(\frac{1}{\mathrm{ii}}\left(\sqrt{29}-5\right)\right),2\pi +{\tan }^{-1}\left(\frac{1}{2}\left(-\sqrt{29}-5\right)\right)$

49.

There are no solutions.

51.

At that place are no solutions.

53.

$0,\frac{2\pi }{3},\frac{\mathrm{iv}\pi }{3}$

55.

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

57.

${\sin }^{-\mathrm{i}}\left(\frac{3}{5}\right),\frac{\pi }{2},\pi -{\sin }^{-1}\left(\frac{3}{5}\right),\frac{3\pi }{\mathrm{two}}$

59.

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

61.

$\frac{\pi }{\mathrm{three}},{\cos }^{-\mathrm{i}}\left(-\frac{\mathrm{three}}{\mathrm{four}}\right),2\pi -{\cos }^{-1}\left(-\frac{\mathrm{three}}{\mathrm{four}}\right),\frac{5\pi }{\mathrm{three}}$

63.

${\cos }^{-\mathrm{ane}}\left(\frac{3}{4}\right),{\cos }^{-\mathrm{one}}\left(-\frac{2}{3}\right),2\pi -{\cos }^{-\mathrm{one}}\left(-\frac{2}{3}\right)$ , $\mathrm{two}\pi -{\cos }^{-\mathrm{ane}}\left(\frac{3}{4}\right)$

65.

$0,\frac{\pi }{2},\pi ,\frac{\mathrm{iii}\pi }{2}$

67.

$\frac{\pi }{\mathrm{three}},{\cos }^{\mathrm{-1}}\left(-\frac{1}{\mathrm{four}}\right),2\pi -{\cos }^{\mathrm{-1}}\left(-\frac{\mathrm{ane}}{4}\right),\frac{5\pi }{\mathrm{iii}}$

69.

There are no solutions.

71.

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

73.

$2\pi \mathrm{one\; thousand}+0.2734,2\pi \mathrm{1000}+\mathrm{ii.8682}$

77.

$0.6694,1.8287,\mathrm{iii.8110},4.9703$

79.

$1.0472,\mathrm{iii.1416},\mathrm{five.2360}$

81.

$0.5326,1.7648,3.6742,4.9064$

83.

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

85.

$\frac{\pi }{2},\frac{3\pi }{\mathrm{two}}$

87.

There are no solutions.

89.

$0,\frac{\pi }{\mathrm{two}},\pi ,\frac{\mathrm{iii}\pi }{2}$

91.

There are no solutions.

7.6 Section Exercises

ane.

Concrete behavior should be periodic, or cyclical.

3.

Since cumulative rainfall is always increasing, a sinusoidal part would non be ideal hither.

v.

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

viii.

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

ten.

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

12.

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

23.

From June xv through November 16

25.

From mean solar day 31 through twenty-four hour period 58

27.

Floods: Apr 16 to July 15. Drought: Oct 16 to Jan 15.

29.

Aamplitude: 8, period: $\frac{1}{\mathrm{iii}},$ frequency: iii Hz

31.

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

33.

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

35.

$P(t)=-33\cos \left(\frac{\pi }{\mathrm{vi}}t\right)+900+{\left(\mathrm{i.07}\right)}^{}t$

37.

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

39.

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

45.

Jump 2 comes to remainder commencement after 7.3 seconds.

47.

234.3 miles, at 72.2°

49.

$y=\mathrm{half\; dozen}{\left(4\right)}^{\mathrm{ten}}+\mathrm{v}\sin \left(\frac{\pi }{\mathrm{ii}}x\right)$

51.

$y=4{(–2)}^{\mathrm{10}}+\mathrm{eight}\sin \left(\frac{\pi }{2}x\right)$

53.

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

Review Exercises

one.

${\sin }^{-1}\left(\frac{\sqrt{3}}{3}\right),\pi -{\sin }^{-1}\left(\frac{\sqrt{3}}{3}\right),\pi +{\sin }^{-\mathrm{ane}}\left(\frac{\sqrt{\mathrm{three}}}{\mathrm{three}}\right),\mathrm{two}\pi -{\sin }^{-1}\left(\frac{\sqrt{3}}{3}\right)$

3.

$\frac{\mathrm{seven}\pi }{6},\frac{11\pi }{\mathrm{vi}}$

5.

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

15.

$\begin{array}{l}\cos \left(4x\right)-\cos \left(3\mathrm{ten}\right)\cos x=\cos \left(2x+\mathrm{two}x\right)-\cos \left(\mathrm{ten}+\mathrm{two}x\right)\cos \mathrm{ten}\hfill \\ =\cos \left(2\mathrm{ten}\right)\cos \left(\mathrm{ii}\mathrm{ten}\right)-\sin \left(2\mathrm{10}\right)\sin \left(\mathrm{two}\mathrm{10}\right)-\cos x\cos \left(\mathrm{two}x\right)\cos x+\sin x\sin \left(2\mathrm{10}\right)\cos x\hfill \\ ={\left({\cos }^{2}x-{\sin }^2\mathrm{10}\right)}^{\mathrm{ii}}-\mathrm{iv}{\cos }^2\mathrm{ten}{\sin }^{\mathrm{ii}}x-{\cos }^{\mathrm{two}}\mathrm{ten}\left({\cos }^{\mathrm{ii}}x-{\sin }^{\mathrm{two}}x\right)+\sin \mathrm{10}\left(\mathrm{ii}\right)\sin x\cos \mathrm{ten}\cos x\hfill \\ ={\left({\cos }^{\mathrm{ii}}\mathrm{10}-{\sin }^{\mathrm{ii}}\mathrm{10}\right)}^2-4{\cos }^{\mathrm{two}}x{\sin }^2\mathrm{ten}-{\cos }^2\mathrm{ten}\left({\cos }^{\mathrm{two}}\mathrm{ten}-{\sin }^{2}x\right)+\mathrm{ii}{\sin }^2\mathrm{ten}{\cos }^{2}x\hfill \\ ={\cos }^{\mathrm{four}}x-2{\cos }^{2}x{\sin }^{2}x+{\sin }^4\mathrm{ten}-\mathrm{four}{\cos }^{2}x{\sin }^{2}x-{\cos }^4\mathrm{10}+{\cos }^{2}x{\sin }^{\mathrm{ii}}x+2{\sin }^2\mathrm{10}{\cos }^{2}x\hfill \\ ={\sin }^4\mathrm{ten}-\mathrm{iv}{\cos }^{2}x{\sin }^{\mathrm{ii}}\mathrm{10}+{\cos }^{2}x{\sin }^2\mathrm{10}\hfill \\ ={\sin }^{2}x\left({\sin }^{\mathrm{ii}}\mathrm{ten}+{\cos }^2\mathrm{10}\right)-\mathrm{iv}{\cos }^2\mathrm{ten}{\sin }^{\mathrm{ii}}x\hfill \\ ={\sin }^{\mathrm{two}}x-4{\cos }^{\mathrm{two}}\mathrm{ten}{\sin }^{2}x\hfill \end{array}$

17.

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

21.

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

25.

$\frac{\sqrt{\mathrm{two}}}{10},\frac{7\sqrt{2}}{10},\frac{\mathrm{one}}{\mathrm{seven}},\frac{3}{5},\frac{\mathrm{iv}}{5},\frac{3}{4}$

27.

$\begin{array}{l}\hfill \\ \cot \mathrm{10}\cos (2x)=\cot x(1-\mathrm{ii}{\sin }^{2}x)\hfill \\ =\cot x-\frac{\cos x}{\sin x}(2){\sin }^2\mathrm{ten}\hfill \\ =-\mathrm{ii}\sin \mathrm{10}\cos \mathrm{10}+\cot \mathrm{10}\hfill \\ =-\sin (\mathrm{two}x)+\cot x\hfill \end{array}$

29.

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

35.

$\frac{1}{2}\left(\sin (6\mathrm{10})+\sin (12x)\right)$

37.

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

39.

$\frac{\mathrm{iii}\pi }{4},\frac{\mathrm{seven}\pi }{\mathrm{four}}$

41.

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

47.

$0.2527,2.8889,4.7124$

49.

$\mathrm{ane.3694}$ , $\mathrm{ane.9106}$ , $4.3726$ , $\mathrm{iv.9137}$

51.

$3\sin \left(\frac{\mathrm{ten}\pi }{2}\right)-\mathrm{ii}$

55.

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

57.

Aamplitude: 3, menstruation: 2, frequency: $\frac{\mathrm{one}}{2}$ Hz

59.

$C(t)=\mathrm{twenty}\sin (2\pi t)+100{(\mathrm{one.4427})}^t$

Practice Exam

11.

$2\text{cos}\left(\mathrm{three}\mathrm{ten}\right)\text{cos}\left(5x\right)$

13.

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

15.

$\frac{\mathrm{iii}}{5},-\frac{4}{\mathrm{v}},-\frac{3}{\mathrm{four}}$

17.

$\begin{array}{cc}\hfill {\text{tan}}^{3}x–\text{tan}x{\text{sec}}^2\mathrm{ten}& =\text{tan}\mathrm{10}({\text{tan}}^{2}x–{\text{sec}}^{\mathrm{ii}}x)\hfill \\ \hfill & =\text{tan}\mathrm{ten}({\text{tan}}^{2}x–(1+{\text{tan}}^{\mathrm{ii}}\mathrm{10}\left)\right)\hfill \\ \hfill & =\text{tan}x({\text{tan}}^{\mathrm{two}}\mathrm{ten}–1–{\text{tan}}^{2}x)\hfill \\ \hfill & =–\text{tan}x=\text{tan}(–\mathrm{ten})=\text{tan}–\mathrm{10})\hfill \end{array}$

19.

$\begin{array}{cc}\hfill \frac{\text{sin}\left(\mathrm{ii}x\right)}{\text{sin}\mathrm{10}}–\frac{\text{cos}\left(2x\right)}{\text{cos}\mathrm{ten}}& =\frac{2\text{sin}\mathrm{10}\text{cos}x}{\text{sin}\mathrm{10}}–\frac{\mathrm{ii}{\text{cos}}^{\mathrm{ii}}x–\mathrm{ane}}{\text{cos}x}\hfill \\ \hfill & =\mathrm{two}\text{cos}x–2\text{cos}\mathrm{ten}+\frac{\mathrm{ane}}{\text{cos}x}\hfill \\ \hfill & =\frac{1}{\text{cos}\mathrm{ten}}=\text{sec}\mathrm{10}=\text{sec}x\hfill \end{array}$

21.

Amplitude: $\frac{1}{\mathrm{four}}$ , period $\frac{\mathrm{i}}{60}$ , frequency: sixty Hz

23.

Aamplitude: $8$ , fast period: $\frac{1}{500}$ , fast frequency: 500 Hz, boring flow: $\frac{1}{10}$ , wearisome frequency: ten Hz

25.

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

Chapter 7 Test Answer Key,

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

Posted by: klinesuffigh.blogspot.com

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