# 9.5 Solving trigonometric equations  (Page 7/10)

 Page 7 / 10

## Algebraic

For the following exercises, find all solutions exactly on the interval $\text{\hspace{0.17em}}0\le \theta <2\pi .$

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

$2\text{\hspace{0.17em}}\mathrm{sin}\text{\hspace{0.17em}}\theta =\sqrt{3}$

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

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

$2\text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}\theta =-\sqrt{2}$

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

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

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

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

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

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

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

${\mathrm{csc}}^{2}x-4=0$

For the following exercises, solve exactly on $\text{\hspace{0.17em}}\left[0,2\pi \right).$

$2\text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}\theta =\sqrt{2}$

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

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

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

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

$2\text{\hspace{0.17em}}\mathrm{sin}\text{\hspace{0.17em}}\theta =-\sqrt{3}$

$2\text{\hspace{0.17em}}\mathrm{sin}\left(3\theta \right)=1$

$\frac{\pi }{18},\frac{5\pi }{18},\frac{13\pi }{18},\frac{17\pi }{18},\frac{25\pi }{18},\frac{29\pi }{18}$

$2\text{\hspace{0.17em}}\mathrm{sin}\left(2\theta \right)=\sqrt{3}$

$2\text{\hspace{0.17em}}\mathrm{cos}\left(3\theta \right)=-\sqrt{2}$

$\frac{3\pi }{12},\frac{5\pi }{12},\frac{11\pi }{12},\frac{13\pi }{12},\frac{19\pi }{12},\frac{21\pi }{12}$

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

$2\text{\hspace{0.17em}}\mathrm{sin}\left(\pi \theta \right)=1$

$\frac{1}{6},\frac{5}{6},\frac{13}{6},\frac{17}{6},\frac{25}{6},\frac{29}{6},\frac{37}{6}$

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

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

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

$0,\frac{\pi }{3},\pi ,\frac{5\pi }{3}$

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

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

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

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

$2\text{\hspace{0.17em}}\mathrm{sin}\left(x\right)\mathrm{cos}\left(x\right)-\mathrm{sin}\left(x\right)+2\text{\hspace{0.17em}}\mathrm{cos}\left(x\right)-1=0$

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

${\mathrm{cos}}^{2}\theta =\frac{1}{2}$

${\mathrm{sec}}^{2}x=1$

$0,\pi$

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

$8\text{\hspace{0.17em}}{\mathrm{sin}}^{2}\left(x\right)+6\text{\hspace{0.17em}}\mathrm{sin}\left(x\right)+1=0$

$\pi -{\mathrm{sin}}^{-1}\left(-\frac{1}{4}\right),\frac{7\pi }{6},\frac{11\pi }{6},2\pi +{\mathrm{sin}}^{-1}\left(-\frac{1}{4}\right)$

${\mathrm{tan}}^{5}\left(x\right)=\mathrm{tan}\left(x\right)$

For the following exercises, solve with the methods shown in this section exactly on the interval $\text{\hspace{0.17em}}\left[0,2\pi \right).$

$\mathrm{sin}\left(3x\right)\mathrm{cos}\left(6x\right)-\mathrm{cos}\left(3x\right)\mathrm{sin}\left(6x\right)=-0.9$

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

$\mathrm{sin}\left(6x\right)\mathrm{cos}\left(11x\right)-\mathrm{cos}\left(6x\right)\mathrm{sin}\left(11x\right)=-0.1$

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

$0$

$6\text{\hspace{0.17em}}\mathrm{sin}\left(2t\right)+9\text{\hspace{0.17em}}\mathrm{sin}\text{\hspace{0.17em}}t=0$

$9\text{\hspace{0.17em}}\mathrm{cos}\left(2\theta \right)=9\text{\hspace{0.17em}}{\mathrm{cos}}^{2}\theta -4$

$\frac{\pi }{6},\frac{5\pi }{6},\frac{7\pi }{6},\frac{11\pi }{6}$

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

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

$\frac{3\pi }{2},\frac{\pi }{6},\frac{5\pi }{6}$

$\mathrm{cos}\left(6x\right)-\mathrm{cos}\left(3x\right)=0$

For the following exercises, solve exactly on the interval $\text{\hspace{0.17em}}\left[0,2\pi \right).\text{\hspace{0.17em}}$ Use the quadratic formula if the equations do not factor.

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

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

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

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

There are no solutions.

$5\text{\hspace{0.17em}}{\mathrm{cos}}^{2}x+3\text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}x-1=0$

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

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

$5\text{\hspace{0.17em}}{\mathrm{sin}}^{2}x+2\text{\hspace{0.17em}}\mathrm{sin}\text{\hspace{0.17em}}x-1=0$

${\mathrm{tan}}^{2}x+5\mathrm{tan}\text{\hspace{0.17em}}x-1=0$

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

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

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

There are no solutions.

For the following exercises, find exact solutions on the interval $\text{\hspace{0.17em}}\left[0,2\pi \right).\text{\hspace{0.17em}}$ Look for opportunities to use trigonometric identities.

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

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

There are no solutions.

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

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

$0,\frac{2\pi }{3},\frac{4\pi }{3}$

$\frac{2\text{\hspace{0.17em}}\mathrm{tan}\text{\hspace{0.17em}}x}{2-{\mathrm{sec}}^{2}x}-{\mathrm{sin}}^{2}x={\mathrm{cos}}^{2}x$

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

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

${\mathrm{sec}}^{2}x=7$

$10\text{\hspace{0.17em}}\mathrm{sin}\text{\hspace{0.17em}}x\text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}x=6\text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}x$

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

$-3\text{\hspace{0.17em}}\mathrm{sin}\text{\hspace{0.17em}}t=15\text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}t\text{\hspace{0.17em}}\mathrm{sin}\text{\hspace{0.17em}}t$

$4\text{\hspace{0.17em}}{\mathrm{cos}}^{2}x-4=15\text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}x$

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

$8\text{\hspace{0.17em}}{\mathrm{sin}}^{2}x+6\text{\hspace{0.17em}}\mathrm{sin}\text{\hspace{0.17em}}x+1=0$

$8\text{\hspace{0.17em}}{\mathrm{cos}}^{2}\theta =3-2\text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}\theta$

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

$6\text{\hspace{0.17em}}{\mathrm{cos}}^{2}x+7\text{\hspace{0.17em}}\mathrm{sin}\text{\hspace{0.17em}}x-8=0$

$12\text{\hspace{0.17em}}{\mathrm{sin}}^{2}t+\mathrm{cos}\text{\hspace{0.17em}}t-6=0$

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

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

${\mathrm{cos}}^{3}t=\mathrm{cos}\text{\hspace{0.17em}}t$

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

## Graphical

For the following exercises, algebraically determine all solutions of the trigonometric equation exactly, then verify the results by graphing the equation and finding the zeros.

$6\text{\hspace{0.17em}}{\mathrm{sin}}^{2}x-5\text{\hspace{0.17em}}\mathrm{sin}\text{\hspace{0.17em}}x+1=0$

$8\text{\hspace{0.17em}}{\mathrm{cos}}^{2}x-2\text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}x-1=0$

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

$100\text{\hspace{0.17em}}{\mathrm{tan}}^{2}x+20\text{\hspace{0.17em}}\mathrm{tan}\text{\hspace{0.17em}}x-3=0$

$2\text{\hspace{0.17em}}{\mathrm{cos}}^{2}x-\mathrm{cos}\text{\hspace{0.17em}}x+15=0$

There are no solutions.

$20\text{\hspace{0.17em}}{\mathrm{sin}}^{2}x-27\text{\hspace{0.17em}}\mathrm{sin}\text{\hspace{0.17em}}x+7=0$

$2\text{\hspace{0.17em}}{\mathrm{tan}}^{2}x+7\text{\hspace{0.17em}}\mathrm{tan}\text{\hspace{0.17em}}x+6=0$

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

$130\text{\hspace{0.17em}}{\mathrm{tan}}^{2}x+69\text{\hspace{0.17em}}\mathrm{tan}\text{\hspace{0.17em}}x-130=0$

## Technology

For the following exercises, use a calculator to find all solutions to four decimal places.

$\mathrm{sin}\text{\hspace{0.17em}}x=0.27$

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

$\mathrm{sin}\text{\hspace{0.17em}}x=-0.55$

$\mathrm{tan}\text{\hspace{0.17em}}x=-0.34$

$\pi k-0.3277$

$\mathrm{cos}\text{\hspace{0.17em}}x=0.71$

For the following exercises, solve the equations algebraically, and then use a calculator to find the values on the interval $\text{\hspace{0.17em}}\left[0,2\pi \right).\text{\hspace{0.17em}}$ Round to four decimal places.

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

$0.6694,1.8287,3.8110,4.9703$

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