Step by step
The question
Question: solve 2 cos x − 1 = 0.
Find every angle from 0° up to 360°.
We want the angles from 0° to 360°: one full turn.
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.
1 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.
2 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!
3 Find the reference angle
From the unit circle: cos 60° = 1/2.
So the reference angle is 60°.
4 Find every angle
Cos is the across value.
The points with the across value we want are on an up-and-down line.
Cos is positive right of the y-axis: quadrants I and IV.
60° and 360° − 60°: x = 60° or 300°.
Careful!
The calculator gives only 60°.
But the line crosses the circle twice.
300° works too.
On the circle: 2 points, so 2 angles.
So x = 60°, 300°.
5 Check
Let's check every answer in the equation.
Each one must give 0.
Every answer works.
So x = 60°, 300°.
Let's look back.
First: remove the − 1.
Then: remove the 2.
Then: find the reference angle.
Then: find every angle.
Last: check.