# 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.

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

$0.6694,1.8287,3.8110,4.9703$

x exposant 4 + 4 x exposant 3 + 8 exposant 2 + 4 x + 1 = 0
x exposent4+4x exposent3+8x exposent2+4x+1=0
HERVE
How can I solve for a domain and a codomains in a given function?
ranges
EDWIN
Thank you I mean range sir.
Oliver
proof for set theory
don't you know?
Inkoom
find to nearest one decimal place of centimeter the length of an arc of circle of radius length 12.5cm and subtending of centeral angle 1.6rad
factoring polynomial
find general solution of the Tanx=-1/root3,secx=2/root3
find general solution of the following equation
Nani
the value of 2 sin square 60 Cos 60
0.75
Lynne
0.75
Inkoom
when can I use sin, cos tan in a giving question
depending on the question
Nicholas
I am a carpenter and I have to cut and assemble a conventional roof line for a new home. The dimensions are: width 30'6" length 40'6". I want a 6 and 12 pitch. The roof is a full hip construction. Give me the L,W and height of rafters for the hip, hip jacks also the length of common jacks.
John
I want to learn the calculations
where can I get indices
I need matrices
Nasasira
hi
Raihany
Hi
Solomon
need help
Raihany
maybe provide us videos
Nasasira
Raihany
Hello
Cromwell
a
Amie
What do you mean by a
Cromwell
nothing. I accidentally press it
Amie
you guys know any app with matrices?
Khay
Ok
Cromwell
Solve the x? x=18+(24-3)=72
x-39=72 x=111
Suraj
Solve the formula for the indicated variable P=b+4a+2c, for b
Need help with this question please
b=-4ac-2c+P
Denisse
b=p-4a-2c
Suddhen
b= p - 4a - 2c
Snr
p=2(2a+C)+b
Suraj
b=p-2(2a+c)
Tapiwa
P=4a+b+2C
COLEMAN
b=P-4a-2c
COLEMAN
like Deadra, show me the step by step order of operation to alive for b
John
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.
The sequence is {1,-1,1-1.....} has