# 9.5 Solving trigonometric equations  (Page 9/10)

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For the following exercises, simplify the equation algebraically as much as possible. Then use a calculator to find the solutions on the interval $\text{\hspace{0.17em}}\left[0,2\pi \right).\text{\hspace{0.17em}}$ Round to four decimal places.

$\sqrt{3}\text{\hspace{0.17em}}{\mathrm{cot}}^{2}x+\mathrm{cot}\text{\hspace{0.17em}}x=1$

${\mathrm{csc}}^{2}x-3\text{\hspace{0.17em}}\mathrm{csc}\text{\hspace{0.17em}}x-4=0$

$0.2527,2.8889,4.7124$

For the following exercises, graph each side of the equation to find the approximate solutions on the interval $\text{\hspace{0.17em}}\left[0,2\pi \right).$

$20\text{\hspace{0.17em}}{\mathrm{cos}}^{2}x+21\text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}x+1=0$

${\mathrm{sec}}^{2}x-2\text{\hspace{0.17em}}\mathrm{sec}\text{\hspace{0.17em}}x=15$

$1.3694,1.9106,4.3726,4.9137$

## Practice test

For the following exercises, simplify the given expression.

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

1

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

$csc\left(\theta \right)\mathrm{cot}\left(\theta \right)\left({\mathrm{sec}}^{2}\theta -1\right)$

$\mathrm{sec}\left(\theta \right)$

${\mathrm{cos}}^{2}\left(\theta \right){\mathrm{sin}}^{2}\left(\theta \right)\left(1+{\mathrm{cot}}^{2}\left(\theta \right)\right)\left(1+{\mathrm{tan}}^{2}\left(\theta \right)\right)$

1

For the following exercises, find the exact value.

$\mathrm{cos}\left(\frac{7\pi }{12}\right)$

$\frac{\sqrt{2}-\sqrt{6}}{4}$

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

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

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

$2\mathrm{sin}\left(\frac{\pi }{4}\right)\mathrm{sin}\left(\frac{\pi }{6}\right)$

$\mathrm{cos}\left(\frac{4\pi }{3}+\theta \right)$

$-\frac{1}{2}\mathrm{cos}\left(\theta \right)-\frac{\sqrt{3}}{2}\mathrm{sin}\left(\theta \right)$

$\mathrm{tan}\left(-\frac{\pi }{4}+\theta \right)$

$\frac{-1+\mathrm{tan}\left(\theta \right)}{1+\mathrm{tan}\left(\theta \right)}$

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

${\mathrm{cos}}^{2}\left(32°\right){\mathrm{tan}}^{2}\left(32°\right)$

$\frac{1-\mathrm{cos}\left({64}^{\circ }\right)}{2}$

$\mathrm{cot}\left(\frac{\theta }{2}\right)$

$±\sqrt{\frac{1+\mathrm{cos}\left(\theta \right)}{1-\mathrm{cos}\left(\theta \right)}}$

For the following exercises, find all exact solutions to the equation on $\text{\hspace{0.17em}}\left[0,2\pi \right).$

${\mathrm{cos}}^{2}x-{\mathrm{sin}}^{2}x-1=0$

$0,\pi$

${\mathrm{cos}}^{2}x=\mathrm{cos}\text{\hspace{0.17em}}x\text{\hspace{0.17em}}4\text{\hspace{0.17em}}{\mathrm{sin}}^{2}x+2\text{\hspace{0.17em}}\mathrm{sin}\text{\hspace{0.17em}}x-3=0$

${\mathrm{sin}}^{-1}\left(\frac{1}{4}\left(\sqrt{13}-1\right)\right),\pi -{\mathrm{sin}}^{-1}\left(\frac{1}{4}\left(\sqrt{13}-1\right)\right)$

$\mathrm{cos}\left(2x\right)+{\mathrm{sin}}^{2}x=0$

$2\text{\hspace{0.17em}}{\mathrm{sin}}^{2}x-\mathrm{sin}\text{\hspace{0.17em}}x=0$

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

Rewrite the expression as a product instead of a sum: $\text{\hspace{0.17em}}\mathrm{cos}\left(2x\right)+\mathrm{cos}\left(-8x\right).$

For the following exercise, rewrite the product as a sum or difference.

$8\mathrm{cos}\left(15x\right)\mathrm{sin}\left(3x\right)$

$4\left[\mathrm{sin}\left(18x\right)-\mathrm{sin}\left(12x\right)\right]$

For the following exercise, rewrite the sum or difference as a product.

$2\left(\mathrm{sin}\left(8\theta \right)-\mathrm{sin}\left(4\theta \right)\right)$

$4\mathrm{sin}\left(2\theta \right)\mathrm{cos}\left(6\theta \right)$

Find all solutions of $\text{\hspace{0.17em}}\mathrm{tan}\left(x\right)-\sqrt{3}=0.$

$\frac{\pi }{3}+k\pi$

Find the solutions of $\text{\hspace{0.17em}}{\mathrm{sec}}^{2}x-2\text{\hspace{0.17em}}\mathrm{sec}\text{\hspace{0.17em}}x=15\text{\hspace{0.17em}}$ on the interval $\text{\hspace{0.17em}}\left[0,2\pi \right)\text{\hspace{0.17em}}$ algebraically; then graph both sides of the equation to determine the answer.

For the following exercises, find all solutions exactly on the interval $\text{\hspace{0.17em}}0\le \theta \le \pi$

$2\text{}\mathrm{cos}\left(\frac{\theta }{2}\right)=1$

${120}^{\circ }$

$\sqrt{3}\text{}\mathrm{cot}\left(y\right)=1$

${60}^{\circ }$

Find $\text{\hspace{0.17em}}\mathrm{sin}\left(2\theta \right),\mathrm{cos}\left(2\theta \right),$ and $\text{\hspace{0.17em}}\mathrm{tan}\left(2\theta \right)\text{\hspace{0.17em}}$ given $\text{\hspace{0.17em}}\mathrm{cot}\text{\hspace{0.17em}}\theta =-\frac{3}{4}\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\theta \text{\hspace{0.17em}}$ is on the interval $\text{\hspace{0.17em}}\left[\frac{\pi }{2},\pi \right].$

$\text{\hspace{0.17em}}-\frac{24}{25},-\frac{7}{25},\frac{24}{7}$

Find $\text{\hspace{0.17em}}\mathrm{sin}\left(\frac{\theta }{2}\right),\mathrm{cos}\left(\frac{\theta }{2}\right),$ and $\text{\hspace{0.17em}}\mathrm{tan}\left(\frac{\theta }{2}\right)\text{\hspace{0.17em}}$ given $\text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}\theta =\frac{7}{25}\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\theta \text{\hspace{0.17em}}$ is in quadrant IV.

Rewrite the expression $\text{\hspace{0.17em}}{\mathrm{sin}}^{4}x\text{\hspace{0.17em}}$ with no powers greater than 1.

$\frac{1}{8}\left(3+\mathrm{cos}\left(4x\right)-4\mathrm{cos}\left(2x\right)\right)$

For the following exercises, prove the identity.

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

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

$\begin{array}{ccc}\hfill \mathrm{sin}\left(3x\right)-\mathrm{cos}x\mathrm{sin}\left(2x\right)& =& \\ \hfill \mathrm{sin}\left(x+2x\right)-\mathrm{cos}x\left(2\mathrm{sin}x\mathrm{cos}x\right)& =& \\ \hfill \mathrm{sin}x\mathrm{cos}\left(2x\right)+\mathrm{sin}\left(2x\right)\mathrm{cos}x-2\mathrm{sin}x{\mathrm{cos}}^{2}x& =& \\ \hfill \mathrm{sin}x\left({\mathrm{cos}}^{2}x-{\mathrm{sin}}^{2}x\right)+2\mathrm{sin}x\mathrm{cos}x\mathrm{cos}x-2\mathrm{sin}x{\mathrm{cos}}^{2}x& =& \\ \hfill \mathrm{sin}x{\mathrm{cos}}^{2}x-{\mathrm{sin}}^{3}+0& =& \\ \hfill {\mathrm{cos}}^{2}x\mathrm{sin}x-{\mathrm{sin}}^{3}x& =& {\mathrm{cos}}^{2}x\mathrm{sin}x-{\mathrm{sin}}^{3}x\hfill \end{array}$

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

Plot the points and find a function of the form $\text{\hspace{0.17em}}y=A\mathrm{cos}\left(Bx+C\right)+D\text{\hspace{0.17em}}$ that fits the given data.

 $x$ $0$ $1$ $2$ $3$ $4$ $5$ $y$ $-2$ $2$ $-2$ $2$ $-2$ $2$

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

The displacement $\text{\hspace{0.17em}}h\left(t\right)\text{\hspace{0.17em}}$ in centimeters of a mass suspended by a spring is modeled by the function $\text{\hspace{0.17em}}h\left(t\right)=\frac{1}{4}\text{\hspace{0.17em}}\mathrm{sin}\left(120\pi t\right),$ where $\text{\hspace{0.17em}}t\text{\hspace{0.17em}}$ is measured in seconds. Find the amplitude, period, and frequency of this displacement.

A woman is standing 300 feet away from a 2000-foot building. If she looks to the top of the building, at what angle above horizontal is she looking? A bored worker looks down at her from the 15 th floor (1500 feet above her). At what angle is he looking down at her? Round to the nearest tenth of a degree.

$81.5°,78.7°$

Two frequencies of sound are played on an instrument governed by the equation $\text{\hspace{0.17em}}n\left(t\right)=8\text{\hspace{0.17em}}\mathrm{cos}\left(20\pi t\right)\mathrm{cos}\left(1000\pi t\right).\text{\hspace{0.17em}}$ What are the period and frequency of the “fast” and “slow” oscillations? What is the amplitude?

The average monthly snowfall in a small village in the Himalayas is 6 inches, with the low of 1 inch occurring in July. Construct a function that models this behavior. During what period is there more than 10 inches of snowfall?

$6+5\text{\hspace{0.17em}}\mathrm{cos}\left(\frac{\pi }{6}\left(1-x\right)\right)\text{\hspace{0.17em}}$ . From November 23 to February 6.

A spring attached to a ceiling is pulled down 20 cm. After 3 seconds, wherein it completes 6 full periods, the amplitude is only 15 cm. Find the function modeling the position of the spring $\text{\hspace{0.17em}}t\text{\hspace{0.17em}}$ seconds after being released. At what time will the spring come to rest? In this case, use 1 cm amplitude as rest.

Water levels near a glacier currently average 9 feet, varying seasonally by 2 inches above and below the average and reaching their highest point in January. Due to global warming, the glacier has begun melting faster than normal. Every year, the water levels rise by a steady 3 inches. Find a function modeling the depth of the water $\text{\hspace{0.17em}}t\text{\hspace{0.17em}}$ months from now. If the docks are 2 feet above current water levels, at what point will the water first rise above the docks?

$D\left(t\right)=2\text{\hspace{0.17em}}\mathrm{cos}\left(\frac{\pi }{6}t\right)+108+\frac{1}{4}t,$ 93.5855 months (or 7.8 years) from now

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