How do you solve cos^2x=3/4? Which solution is correct?

cos^2x=3/4
This is already covered here:
https://socratic.org/questions/how-do-you-solve-cos-2x-3-4
But my solution:
1. cosx=(sqrt3)/2
2. cosx=-(sqrt3)/2
So:
1. arccos((sqrt3)/2)+2pik=pi/6+2pik
-arccos((sqrt3)/2)+2pik=-pi/6+2pik
2. arccos(-(sqrt3)/2)+2pik=pi-arccos((sqrt3)/2)+2pik= pi-pi/6+2pik=(5pi)/6+pik
-arccos(-(sqrt3)/2)+2pik=pi+arccos((sqrt3)/2)+2pik=pi+pi/6+2pik=(7pi)/6+2pik
Which solution is correct? My or here:
https://socratic.org/questions/how-do-you-solve-cos-2x-3-4

3 Answers

You should consider two cases as follows

(1) \cos x=\sqrt3/2=\cos(\pi/6)

x=2k\pi\pm\pi/6

Where k is any integer

(2) \cos x=-\sqrt3/2=\cos({5\pi}/6)

x=2k\pi\pm{5\pi}/6

Where k is any integer

Method-2 In general,

\cos^2\theta=\cos^2\alpha has solutions

\theta=n\pi\pm\alpha

Where, n is any integer

Hence, the given trig. equation has following solutions

\cos^2x=3/4

\cos^2x=(\sqrt3/2)^2

\cos^2x=(\cos(\pi/6))^2

\cos^2x=\cos^2(\pi/6)

x=n\pi\pm \pi/6

Where, n is any integer i.e. n=0, \pm1, \pm2, \pm3, \ldots

Jul 4, 2018

color(blue)(x=2kpi+-pi/6 ,kinZZ or color(blue)(x=2kpi+-(5pi)/6 ,kinZZ

Explanation:

Here, cos^2x=3/4=>cosx=+-sqrt3/2
"If" cosx=a, |a|<=1 ,"then the general solution of eqn. is :"

color(blue)(x=2kpi+-alpha, kinZZandalpha=arc cos(a)
(i)cosx=sqrt3/2 >0
=>alpha=arc cos(sqrt3/2)=arc cos(cos(pi/6))=pi/6
color(blue)(x=2kpi+-pi/6 ,kinZZ
(i)cosx=-sqrt3/2 < 0
=>alpha=arc cos(-sqrt3/2)=pi-arc cos(sqrt3/2)=pi-pi/6=(5pi)/6
color(blue)(x=2kpi+-(5pi)/6 ,kinZZ
......................................................................................................
You have to think on red color.

Compare your answer with red color to correct your answer.

2. arc cos(-sqrt3/2)+2pik
=pi-arccos(sqrt3/2)+2pik=pi-pi/6+2pik=(5pi)/6+color(red)(pik

So, replace , color(red)(pik to2pik ,k inZZ

and arc cos(-sqrt3/2)=pi-arc cos(sqrt3/2)

=>color(red)(-){arc cos(-sqrt3/2)}=color(red)(-){pi-arc cos(sqrt3/2)}
=color(red)(-pi+arccos(sqrt3/2)=-pi+pi/6=(-6pi+pi)/6=-(5pi)/6

So, Replace , color(red)((7pi)/6 to-(5pi)/6

Also, remember to write : color(red)(kinZZto it is necessary.

Jul 5, 2018

x=pi/6+-pin
x=(5pi)/6+-pin
Where n is an element of all integers
n∈Z

Explanation:

An easy way to approach this would be to use the unit circle instead of using inverse trig:

cosx=+-sqrt3/2

Now we know using the unit circle, the solutions are pi/6, (5pi)/6, (7pi)/6, (11pi)/6

So the easiest way to write the solution set including all solutions would be:

x=pi/6+-pin
x=(5pi)/6+-pin

Where n is an element of all integers
n∈Z

graph{(cosx)^2-3/4 [-10, 10, -5, 5]}