Step by step
The question
Question: solve 4 cos² x = 1.
Find every angle from 0° up to 360°.
We want the angles from 0° to 360°: one full turn.
1 Remove the 4
4 cos² x means 4 times cos² x.
We want just cos² x.
To remove times 4, we divide by 4.
We do it on both sides.
Why both sides?
The = sign means both sides are the same amount.
If we change only one side, they are not the same any more.
4 divided by 4 is 1.
So just cos² x is left.
So cos² x = 1/4!
2 Square root of both sides
cos x squared is 1/4.
So cos x is the square root of 1/4, or minus it.
Both work.
The root of a fraction: root of the top over root of the bottom.
Careful!
A square root gives two answers: plus and minus.
cos x = −1/2 works too.
So cos x = 1/2 or cos x = −1/2.
3 Find the reference angle
From the unit circle: cos 60° = 1/2.
So the reference angle is 60°.
The minus only changes which quadrants.
4 Find every angle
Cos is the across value.
The points with the across value we want are on an up-and-down line, one line for each value.
Cos is positive right of the y-axis: quadrants I and IV.
60° and 360° − 60°: x = 60° or 300°.
Cos is negative left of the y-axis: quadrants II and III.
180° − 60° and 180° + 60°: x = 120° or 240°.
On the circle: 4 points, so 4 angles.
So x = 60°, 120°, 240°, 300°.
5 Check
Let's check every answer in the equation.
Each one must give 1.
Every answer works.
So x = 60°, 120°, 240°, 300°.
Let's look back.
First: remove the 4.
Then: square root of both sides.
Then: find the reference angle.
Then: find every angle.
Last: check.