Step by step
The question
Question: solve 2 cos² x − cos x = 0.
Find every angle from 0° up to 360°.
We want the angles from 0° to 360°: one full turn.
Careful!
Do not divide by cos x!
That throws away cos x = 0, and its answers.
We take cos x out instead.
1 Take out the common factor
Both parts have cos x in them.
We take it out, like a common factor.
Check the first part: cos x times 2 cos x is 2 cos squared x.
2 Set each factor to 0
Two things times each other make 0.
That only happens when one of them is 0.
So cos x = 0, or...
...
2 cos x − 1 = 0.
We want cos x by itself.
Two things are stuck to cos x: times 2 and − 1.
The − 1 is on the outside, so it comes off first.
Then the times 2.
3 Remove the − 1
To remove − 1, we add 1.
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.
− 1 and + 1 make zero.
Cross them out.
0 + 1 = 1.
Now we have 2 cos x = 1.
4 Remove the 2
2 cos x means 2 times cos x.
We want just cos x.
To remove times 2, we divide by 2.
We do it on both sides, so they stay equal.
2 divided by 2 is 1.
So just cos x is left.
So cos x = 1/2!
5 Find the reference angle
From the unit circle: cos 60° = 1/2.
So the reference angle is 60°.
6 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 0 straight up and straight down.
x = 90° or 270°.
Cos is positive right of the y-axis: quadrants I and IV.
60° and 360° − 60°: x = 60° or 300°.
On the circle: 4 points, so 4 angles.
So x = 60°, 90°, 270°, 300°.
7 Check
Let's check every answer in the equation.
Each one must give 0.
Every answer works.
So x = 60°, 90°, 270°, 300°.
Let's look back.
First: take out the common factor.
Then: set each factor to 0.
Then: remove the − 1.
Then: remove the 2.
Then: find the reference angle.
Then: find every angle.
Last: check.