9.3 Double-angle, half-angle, and reduction formulas  (Page 5/8)

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If $\text{\hspace{0.17em}}\mathrm{tan}\text{\hspace{0.17em}}x=-8,$ and $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ is in quadrant IV.

For the following exercises, find the values of the six trigonometric functions if the conditions provided hold.

$\mathrm{cos}\left(2\theta \right)=\frac{3}{5}\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}90°\le \theta \le 180°$

$\mathrm{cos}\text{\hspace{0.17em}}\theta =-\frac{2\sqrt{5}}{5},\mathrm{sin}\text{\hspace{0.17em}}\theta =\frac{\sqrt{5}}{5},\mathrm{tan}\text{\hspace{0.17em}}\theta =-\frac{1}{2},\mathrm{csc}\text{\hspace{0.17em}}\theta =\sqrt{5},\mathrm{sec}\text{\hspace{0.17em}}\theta =-\frac{\sqrt{5}}{2},\mathrm{cot}\text{\hspace{0.17em}}\theta =-2$

$\mathrm{cos}\left(2\theta \right)=\frac{1}{\sqrt{2}}\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}180°\le \theta \le 270°$

For the following exercises, simplify to one trigonometric expression.

$2\text{\hspace{0.17em}}\mathrm{sin}\left(\frac{\pi }{4}\right)\text{\hspace{0.17em}}2\text{\hspace{0.17em}}\mathrm{cos}\left(\frac{\pi }{4}\right)$

$2\text{\hspace{0.17em}}\mathrm{sin}\left(\frac{\pi }{2}\right)$

$4\text{\hspace{0.17em}}\mathrm{sin}\left(\frac{\pi }{8}\right)\text{\hspace{0.17em}}\mathrm{cos}\left(\frac{\pi }{8}\right)$

For the following exercises, find the exact value using half-angle formulas.

$\text{\hspace{0.17em}}\mathrm{sin}\left(\frac{\pi }{8}\right)\text{\hspace{0.17em}}$

$\frac{\sqrt{2-\sqrt{2}}}{2}$

$\mathrm{cos}\left(-\frac{11\pi }{12}\right)$

$\mathrm{sin}\left(\frac{11\pi }{12}\right)$

$\frac{\sqrt{2-\sqrt{3}}}{2}$

$\mathrm{cos}\left(\frac{7\pi }{8}\right)$

$\mathrm{tan}\left(\frac{5\pi }{12}\right)$

$2+\sqrt{3}$

$\mathrm{tan}\left(-\frac{3\pi }{12}\right)$

$\mathrm{tan}\left(-\frac{3\pi }{8}\right)$

$-1-\sqrt{2}$

For the following exercises, find the exact values of a) $\text{\hspace{0.17em}}\mathrm{sin}\left(\frac{x}{2}\right),$ b) $\text{\hspace{0.17em}}\mathrm{cos}\left(\frac{x}{2}\right),$ and c) $\text{\hspace{0.17em}}\mathrm{tan}\left(\frac{x}{2}\right)$ without solving for $\text{\hspace{0.17em}}x.$

If $\text{\hspace{0.17em}}\mathrm{tan}\text{\hspace{0.17em}}x=-\frac{4}{3},$ and $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ is in quadrant IV.

If $\text{\hspace{0.17em}}\mathrm{sin}\text{\hspace{0.17em}}x=-\frac{12}{13},$ and $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ is in quadrant III.

a) $\text{\hspace{0.17em}}\frac{3\sqrt{13}}{13}\text{\hspace{0.17em}}$ b) $\text{\hspace{0.17em}}-\frac{2\sqrt{13}}{13}\text{\hspace{0.17em}}$ c) $\text{\hspace{0.17em}}-\frac{3}{2}\text{\hspace{0.17em}}$

If $\text{\hspace{0.17em}}\mathrm{csc}\text{\hspace{0.17em}}x=7,$ and $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ is in quadrant II.

If $\text{\hspace{0.17em}}\mathrm{sec}\text{\hspace{0.17em}}x=-4,$ and $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ is in quadrant II.

a) $\text{\hspace{0.17em}}\frac{\sqrt{10}}{4}\text{\hspace{0.17em}}$ b) $\text{\hspace{0.17em}}\frac{\sqrt{6}}{4}\text{\hspace{0.17em}}$ c) $\text{\hspace{0.17em}}\frac{\sqrt{15}}{3}\text{\hspace{0.17em}}$

For the following exercises, use [link] to find the requested half and double angles.

Find $\text{\hspace{0.17em}}\mathrm{sin}\left(2\theta \right),\mathrm{cos}\left(2\theta \right),$ and $\text{\hspace{0.17em}}\mathrm{tan}\left(2\theta \right).$

Find $\text{\hspace{0.17em}}\mathrm{sin}\left(2\alpha \right),\mathrm{cos}\left(2\alpha \right),$ and $\text{\hspace{0.17em}}\mathrm{tan}\left(2\alpha \right).$

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

Find $\text{\hspace{0.17em}}\mathrm{sin}\left(\frac{\theta }{2}\right),\mathrm{cos}\left(\frac{\theta }{2}\right),$ and $\text{\hspace{0.17em}}\mathrm{tan}\left(\frac{\theta }{2}\right).$

Find $\text{\hspace{0.17em}}\mathrm{sin}\left(\frac{\alpha }{2}\right),\mathrm{cos}\left(\frac{\alpha }{2}\right),$ and $\text{\hspace{0.17em}}\mathrm{tan}\left(\frac{\alpha }{2}\right).$

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

For the following exercises, simplify each expression. Do not evaluate.

${\mathrm{cos}}^{2}\left(28°\right)-{\mathrm{sin}}^{2}\left(28°\right)$

$2{\mathrm{cos}}^{2}\left(37°\right)-1$

$\mathrm{cos}\left(74°\right)$

$1-2\text{\hspace{0.17em}}{\mathrm{sin}}^{2}\left(17°\right)$

${\mathrm{cos}}^{2}\left(9x\right)-{\mathrm{sin}}^{2}\left(9x\right)$

$\mathrm{cos}\left(18x\right)$

$4\text{\hspace{0.17em}}\mathrm{sin}\left(8x\right)\text{\hspace{0.17em}}\mathrm{cos}\left(8x\right)$

$6\text{\hspace{0.17em}}\mathrm{sin}\left(5x\right)\text{\hspace{0.17em}}\mathrm{cos}\left(5x\right)$

$3\mathrm{sin}\left(10x\right)$

For the following exercises, prove the given identity.

${\left(\mathrm{sin}\text{\hspace{0.17em}}t-\mathrm{cos}\text{\hspace{0.17em}}t\right)}^{2}=1-\mathrm{sin}\left(2t\right)$

$\mathrm{sin}\left(2x\right)=-2\text{\hspace{0.17em}}\mathrm{sin}\left(-x\right)\text{\hspace{0.17em}}\mathrm{cos}\left(-x\right)$

$-2\text{\hspace{0.17em}}\mathrm{sin}\left(-x\right)\mathrm{cos}\left(-x\right)=-2\left(-\mathrm{sin}\left(x\right)\mathrm{cos}\left(x\right)\right)=\mathrm{sin}\left(2x\right)$

$\mathrm{cot}\text{\hspace{0.17em}}x-\mathrm{tan}\text{\hspace{0.17em}}x=2\text{\hspace{0.17em}}\mathrm{cot}\left(2x\right)$

$\frac{\mathrm{sin}\left(2\theta \right)}{1+\mathrm{cos}\left(2\theta \right)}{\mathrm{tan}}^{2}\theta =\mathrm{tan}\text{\hspace{0.17em}}\theta$

$\begin{array}{ccc}\hfill \frac{\mathrm{sin}\left(2\theta \right)}{1+\mathrm{cos}\left(2\theta \right)}{\mathrm{tan}}^{2}\theta & =& \frac{2\mathrm{sin}\left(\theta \right)\mathrm{cos}\left(\theta \right)}{1+{\mathrm{cos}}^{2}\theta -{\mathrm{sin}}^{2}\theta }{\mathrm{tan}}^{2}\theta =\hfill \\ \hfill \frac{2\mathrm{sin}\left(\theta \right)\mathrm{cos}\left(\theta \right)}{2{\mathrm{cos}}^{2}\theta }{\mathrm{tan}}^{2}\theta & =& \frac{\mathrm{sin}\left(\theta \right)}{\mathrm{cos}\text{\hspace{0.17em}}\theta }{\mathrm{tan}}^{2}\theta =\hfill \\ \hfill \mathrm{cot}\left(\theta \right){\mathrm{tan}}^{2}\theta & =& \mathrm{tan}\text{\hspace{0.17em}}\theta \hfill \end{array}$

For the following exercises, rewrite the expression with an exponent no higher than 1.

${\mathrm{cos}}^{2}\left(5x\right)$

${\mathrm{cos}}^{2}\left(6x\right)$

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

${\mathrm{sin}}^{4}\left(8x\right)$

${\mathrm{sin}}^{4}\left(3x\right)$

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

${\mathrm{cos}}^{2}x{\text{\hspace{0.17em}}\mathrm{sin}}^{4}x$

${\mathrm{cos}}^{4}x{\text{\hspace{0.17em}}\mathrm{sin}}^{2}x$

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

${\mathrm{tan}}^{2}x{\text{\hspace{0.17em}}\mathrm{sin}}^{2}x$

Technology

For the following exercises, reduce the equations to powers of one, and then check the answer graphically.

${\mathrm{tan}}^{4}x$

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

${\mathrm{sin}}^{2}\left(2x\right)$

${\mathrm{sin}}^{2}x{\text{\hspace{0.17em}}\mathrm{cos}}^{2}x$

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

${\mathrm{tan}}^{2}x\text{\hspace{0.17em}}\mathrm{sin}\text{\hspace{0.17em}}x$

${\mathrm{tan}}^{4}x{\text{\hspace{0.17em}}\mathrm{cos}}^{2}x$

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

${\mathrm{cos}}^{2}x\text{\hspace{0.17em}}\mathrm{sin}\left(2x\right)$

${\mathrm{cos}}^{2}\left(2x\right)\mathrm{sin}\text{\hspace{0.17em}}x$

$\frac{\left(1+\mathrm{cos}\left(4x\right)\right)\mathrm{sin}\text{\hspace{0.17em}}x}{2}$

${\mathrm{tan}}^{2}\left(\frac{x}{2}\right)\text{\hspace{0.17em}}\mathrm{sin}\text{\hspace{0.17em}}x$

For the following exercises, algebraically find an equivalent function, only in terms of $\text{\hspace{0.17em}}\mathrm{sin}\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ and/or $\text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}x,$ and then check the answer by graphing both functions.

$\mathrm{sin}\left(4x\right)$

$4\mathrm{sin}\text{\hspace{0.17em}}x\mathrm{cos}\text{\hspace{0.17em}}x\left({\mathrm{cos}}^{2}x-{\mathrm{sin}}^{2}x\right)$

$\mathrm{cos}\left(4x\right)$

Extensions

For the following exercises, prove the identities.

$\mathrm{sin}\left(2x\right)=\frac{2\text{\hspace{0.17em}}\mathrm{tan}\text{\hspace{0.17em}}x}{1+{\mathrm{tan}}^{2}x}$

$\begin{array}{l}\frac{2\mathrm{tan}\text{\hspace{0.17em}}x}{1+{\mathrm{tan}}^{2}x}=\frac{\frac{2\mathrm{sin}\text{\hspace{0.17em}}x}{\mathrm{cos}\text{\hspace{0.17em}}x}}{1+\frac{{\mathrm{sin}}^{2}x}{{\mathrm{cos}}^{2}x}}=\frac{\frac{2\mathrm{sin}\text{\hspace{0.17em}}x}{\mathrm{cos}\text{\hspace{0.17em}}x}}{\frac{{\mathrm{cos}}^{2}x+{\mathrm{sin}}^{2}x}{{\mathrm{cos}}^{2}x}}=\\ \frac{2\mathrm{sin}\text{\hspace{0.17em}}x}{\mathrm{cos}\text{\hspace{0.17em}}x}.\frac{{\mathrm{cos}}^{2}x}{1}=2\mathrm{sin}\text{\hspace{0.17em}}x\mathrm{cos}\text{\hspace{0.17em}}x=\mathrm{sin}\left(2x\right)\end{array}$

$\mathrm{cos}\left(2\alpha \right)=\frac{1-{\mathrm{tan}}^{2}\alpha }{1+{\mathrm{tan}}^{2}\alpha }$

$\mathrm{tan}\left(2x\right)=\frac{2\text{\hspace{0.17em}}\mathrm{sin}\text{\hspace{0.17em}}x\text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}x}{2{\mathrm{cos}}^{2}x-1}$

$\frac{2\mathrm{sin}\text{\hspace{0.17em}}x\mathrm{cos}\text{\hspace{0.17em}}x}{2{\mathrm{cos}}^{2}x-1}=\frac{\mathrm{sin}\left(2x\right)}{\mathrm{cos}\left(2x\right)}=\mathrm{tan}\left(2x\right)$

${\left({\mathrm{sin}}^{2}x-1\right)}^{2}=\mathrm{cos}\left(2x\right)+{\mathrm{sin}}^{4}x$

$\mathrm{sin}\left(3x\right)=3\text{\hspace{0.17em}}\mathrm{sin}\text{\hspace{0.17em}}x\text{\hspace{0.17em}}{\mathrm{cos}}^{2}x-{\mathrm{sin}}^{3}x$

$\begin{array}{ccc}\hfill \mathrm{sin}\left(x+2x\right)& =& \mathrm{sin}\text{\hspace{0.17em}}x\mathrm{cos}\left(2x\right)+\mathrm{sin}\left(2x\right)\mathrm{cos}\text{\hspace{0.17em}}x\hfill \\ & =& \mathrm{sin}\text{\hspace{0.17em}}x\left({\mathrm{cos}}^{2}x-{\mathrm{sin}}^{2}x\right)+2\mathrm{sin}\text{\hspace{0.17em}}x\mathrm{cos}\text{\hspace{0.17em}}x\mathrm{cos}\text{\hspace{0.17em}}x\hfill \\ & =& \mathrm{sin}\text{\hspace{0.17em}}x{\mathrm{cos}}^{2}x-{\mathrm{sin}}^{3}x+2\mathrm{sin}\text{\hspace{0.17em}}x{\mathrm{cos}}^{2}x\hfill \\ & =& 3\mathrm{sin}\text{\hspace{0.17em}}x{\mathrm{cos}}^{2}x-{\mathrm{sin}}^{3}x\hfill \end{array}$

$\mathrm{cos}\left(3x\right)={\mathrm{cos}}^{3}x-3{\mathrm{sin}}^{2}x\text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}x$

$\frac{1+\mathrm{cos}\left(2t\right)}{\mathrm{sin}\left(2t\right)-\mathrm{cos}\text{\hspace{0.17em}}t}=\frac{2\text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}t}{2\text{\hspace{0.17em}}\mathrm{sin}\text{\hspace{0.17em}}t-1}$

$\begin{array}{ccc}\hfill \frac{1+\mathrm{cos}\left(2t\right)}{\mathrm{sin}\left(2t\right)-\mathrm{cos}t}& =& \frac{1+2{\mathrm{cos}}^{2}t-1}{2\mathrm{sin}t\mathrm{cos}t-\mathrm{cos}t}\hfill \\ & =& \frac{2{\mathrm{cos}}^{2}t}{\mathrm{cos}t\left(2\mathrm{sin}t-1\right)}\hfill \\ & =& \frac{2\mathrm{cos}t}{2\mathrm{sin}t-1}\hfill \end{array}$

$\mathrm{sin}\left(16x\right)=16\text{\hspace{0.17em}}\mathrm{sin}\text{\hspace{0.17em}}x\text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}x\text{\hspace{0.17em}}\mathrm{cos}\left(2x\right)\mathrm{cos}\left(4x\right)\mathrm{cos}\left(8x\right)$

$\mathrm{cos}\left(16x\right)=\left({\mathrm{cos}}^{2}\left(4x\right)-{\mathrm{sin}}^{2}\left(4x\right)-\mathrm{sin}\left(8x\right)\right)\left({\mathrm{cos}}^{2}\left(4x\right)-{\mathrm{sin}}^{2}\left(4x\right)+\mathrm{sin}\left(8x\right)\right)$

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

A laser rangefinder is locked on a comet approaching Earth. The distance g(x), in kilometers, of the comet after x days, for x in the interval 0 to 30 days, is given by g(x)=250,000csc(π30x). Graph g(x) on the interval [0, 35]. Evaluate g(5)  and interpret the information. What is the minimum distance between the comet and Earth? When does this occur? To which constant in the equation does this correspond? Find and discuss the meaning of any vertical asymptotes.
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