9.5 Solving trigonometric equations  (Page 9/10)

 Page 9 / 10

${\mathrm{csc}}^{2}t=3$

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

${\mathrm{cos}}^{2}x=\frac{1}{4}$

$2\text{\hspace{0.17em}}\mathrm{sin}\text{\hspace{0.17em}}\theta =-1$

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

$\mathrm{tan}\text{\hspace{0.17em}}x\text{\hspace{0.17em}}\mathrm{sin}\text{\hspace{0.17em}}x+\mathrm{sin}\left(-x\right)=0$

$9\text{\hspace{0.17em}}\mathrm{sin}\text{\hspace{0.17em}}\omega -2=4\text{\hspace{0.17em}}{\mathrm{sin}}^{2}\omega$

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

$1-2\text{\hspace{0.17em}}\mathrm{tan}\left(\omega \right)={\mathrm{tan}}^{2}\left(\omega \right)$

For the following exercises, use basic identities to simplify the expression.

$\mathrm{sec}\text{\hspace{0.17em}}x\text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}x+\mathrm{cos}\text{\hspace{0.17em}}x-\frac{1}{\mathrm{sec}\text{\hspace{0.17em}}x}$

$1$

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

For the following exercises, determine if the given identities are equivalent.

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

Yes

${\mathrm{tan}}^{3}x\text{\hspace{0.17em}}{\mathrm{csc}}^{2}x\text{\hspace{0.17em}}{\mathrm{cot}}^{2}x\text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}x\text{\hspace{0.17em}}\mathrm{sin}\text{\hspace{0.17em}}x=1$

Sum and Difference Identities

For the following exercises, find the exact value.

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

$-2-\sqrt{3}$

$\mathrm{cos}\left(\frac{25\pi }{12}\right)$

$\mathrm{sin}\left(70°\right)\mathrm{cos}\left(25°\right)-\mathrm{cos}\left(70°\right)\mathrm{sin}\left(25°\right)$

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

$\mathrm{cos}\left(83°\right)\mathrm{cos}\left(23°\right)+\mathrm{sin}\left(83°\right)\mathrm{sin}\left(23°\right)$

For the following exercises, prove the identity.

$\mathrm{cos}\left(4x\right)-\mathrm{cos}\left(3x\right)\mathrm{cos}x={\mathrm{sin}}^{2}x-4\text{\hspace{0.17em}}{\mathrm{cos}}^{2}x\text{\hspace{0.17em}}{\mathrm{sin}}^{2}x$

$\begin{array}{ccc}\hfill \mathrm{cos}\left(4x\right)-\mathrm{cos}\left(3x\right)\mathrm{cos}x& =& \mathrm{cos}\left(2x+2x\right)-\mathrm{cos}\left(x+2x\right)\mathrm{cos}\text{\hspace{0.17em}}x\hfill \\ & =& \mathrm{cos}\left(2x\right)\mathrm{cos}\left(2x\right)-\mathrm{sin}\left(2x\right)\mathrm{sin}\left(2x\right)-\mathrm{cos}\text{\hspace{0.17em}}x\text{\hspace{0.17em}}\mathrm{cos}\left(2x\right)\mathrm{cos}\text{\hspace{0.17em}}x+\mathrm{sin}\text{\hspace{0.17em}}x\text{\hspace{0.17em}}\mathrm{sin}\left(2x\right)\mathrm{cos}\text{\hspace{0.17em}}x\hfill \\ & =& {\left({\mathrm{cos}}^{2}x-{\mathrm{sin}}^{2}x\right)}^{2}-4\text{\hspace{0.17em}}{\mathrm{cos}}^{2}x\text{\hspace{0.17em}}{\mathrm{sin}}^{2}x-{\mathrm{cos}}^{2}x\left({\mathrm{cos}}^{2}x-{\mathrm{sin}}^{2}x\right)+\mathrm{sin}\text{\hspace{0.17em}}x\left(2\right)\mathrm{sin}\text{\hspace{0.17em}}x\text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}x\text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}x\hfill \\ & =& {\left({\mathrm{cos}}^{2}x-{\mathrm{sin}}^{2}x\right)}^{2}-4\text{\hspace{0.17em}}{\mathrm{cos}}^{2}x\text{\hspace{0.17em}}{\mathrm{sin}}^{2}x-{\mathrm{cos}}^{2}x\left({\mathrm{cos}}^{2}x-{\mathrm{sin}}^{2}x\right)+2\text{\hspace{0.17em}}{\mathrm{sin}}^{2}x\text{\hspace{0.17em}}{\mathrm{cos}}^{2}x\hfill \\ & =& {\mathrm{cos}}^{4}x-2\text{\hspace{0.17em}}{\mathrm{cos}}^{2}x\text{\hspace{0.17em}}{\mathrm{sin}}^{2}x+{\mathrm{sin}}^{4}x-4\text{\hspace{0.17em}}{\mathrm{cos}}^{2}x\text{\hspace{0.17em}}{\mathrm{sin}}^{2}x-{\mathrm{cos}}^{4}x+{\mathrm{cos}}^{2}x\text{\hspace{0.17em}}{\mathrm{sin}}^{2}x+2\text{\hspace{0.17em}}{\mathrm{sin}}^{2}x\text{\hspace{0.17em}}{\mathrm{cos}}^{2}x\hfill \\ & =& {\mathrm{sin}}^{4}x-4\text{\hspace{0.17em}}{\mathrm{cos}}^{2}x\text{\hspace{0.17em}}{\mathrm{sin}}^{2}x+{\mathrm{cos}}^{2}x\text{\hspace{0.17em}}{\mathrm{sin}}^{2}x\hfill \\ & =& {\mathrm{sin}}^{2}x\left({\mathrm{sin}}^{2}x+{\mathrm{cos}}^{2}x\right)-4\text{\hspace{0.17em}}{\mathrm{cos}}^{2}x\text{\hspace{0.17em}}{\mathrm{sin}}^{2}x\hfill \\ & =& {\mathrm{sin}}^{2}x-4\text{\hspace{0.17em}}{\mathrm{cos}}^{2}x{\mathrm{sin}}^{2}x\hfill \end{array}$

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

For the following exercise, simplify the expression.

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

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

For the following exercises, find the exact value.

$\mathrm{cos}\left({\mathrm{sin}}^{-1}\left(0\right)-{\mathrm{cos}}^{-1}\left(\frac{1}{2}\right)\right)$

$\mathrm{tan}\left({\mathrm{sin}}^{-1}\left(0\right)+{\mathrm{sin}}^{-1}\left(\frac{1}{2}\right)\right)$

$\frac{\sqrt{3}}{3}$

Double-Angle, Half-Angle, and Reduction Formulas

For the following exercises, find the exact value.

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 $\text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}\theta =-\frac{1}{3}\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\theta \text{\hspace{0.17em}}$ is in the interval $\text{\hspace{0.17em}}\left[\frac{\pi }{2},\pi \right].$

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 $\text{\hspace{0.17em}}\mathrm{sec}\text{\hspace{0.17em}}\theta =-\frac{5}{3}\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\theta \text{\hspace{0.17em}}$ is in the interval $\text{\hspace{0.17em}}\left[\frac{\pi }{2},\pi \right].$

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

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

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

$\sqrt{2\left(2+\sqrt{2}\right)}$

For the following exercises, use [link] to find the desired quantities.

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

For the following exercises, prove the identity.

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

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

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

For the following exercises, rewrite the expression with no powers.

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

${\mathrm{tan}}^{2}x\text{\hspace{0.17em}}{\mathrm{sin}}^{3}x$

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

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

For the following exercises, evaluate the product for the given expression using a sum or difference of two functions. Write the exact answer.

$\mathrm{cos}\left(\frac{\pi }{3}\right)\text{\hspace{0.17em}}\mathrm{sin}\left(\frac{\pi }{4}\right)$

$2\text{\hspace{0.17em}}\mathrm{sin}\left(\frac{2\pi }{3}\right)\text{\hspace{0.17em}}\mathrm{sin}\left(\frac{5\pi }{6}\right)$

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

$2\text{\hspace{0.17em}}\mathrm{cos}\left(\frac{\pi }{5}\right)\text{\hspace{0.17em}}\mathrm{cos}\left(\frac{\pi }{3}\right)$

For the following exercises, evaluate the sum by using a product formula. Write the exact answer.

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

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

$\mathrm{cos}\left(\frac{5\pi }{12}\right)+\mathrm{cos}\left(\frac{7\pi }{12}\right)$

For the following exercises, change the functions from a product to a sum or a sum to a product.

$\mathrm{sin}\left(9x\right)\mathrm{cos}\left(3x\right)$

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

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

$\mathrm{sin}\left(11x\right)+\mathrm{sin}\left(2x\right)$

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

$\mathrm{cos}\left(6x\right)+\mathrm{cos}\left(5x\right)$

Solving Trigonometric Equations

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

$\mathrm{tan}\text{\hspace{0.17em}}x+1=0$

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

$2\text{\hspace{0.17em}}\mathrm{sin}\left(2x\right)+\sqrt{2}=0$

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

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

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

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

$2\text{\hspace{0.17em}}{\mathrm{sin}}^{2}x+5\text{\hspace{0.17em}}\mathrm{sin}\text{\hspace{0.17em}}x+3=0$

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

$\mathrm{cos}\text{\hspace{0.17em}}x-5\text{\hspace{0.17em}}\mathrm{sin}\left(2x\right)=0$

$\frac{1}{{\mathrm{sec}}^{2}x}+2+{\mathrm{sin}}^{2}x+4\text{\hspace{0.17em}}{\mathrm{cos}}^{2}x=0$

No solution

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