Step by step
The question
Question: solve 2 sin² x = sin x.
Find every angle from 0° up to 360°.
We want the angles from 0° to 360°: one full turn.
1 Get 0 on one side
Everything goes on the left.
We take away sin x on both sides, so they stay equal.
Careful!
Do not divide by sin x!
That throws away sin x = 0, and its answers.
We take sin x out instead.
2 Take out the common factor
Both parts have sin x in them.
We take it out, like a common factor.
Check the first part: sin x times 2 sin x is 2 sin squared x.
3 Set each factor to 0
Two things times each other make 0.
That only happens when one of them is 0.
So sin x = 0, or...
...
2 sin x − 1 = 0.
We want sin x by itself.
Two things are stuck to sin x: times 2 and − 1.
The − 1 is on the outside, so it comes off first.
Then the times 2.
4 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 sin x = 1.
5 Remove the 2
2 sin x means 2 times sin x.
We want just sin 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 sin x is left.
So sin x = 1/2!
6 Find the reference angle
From the unit circle: sin 30° = 1/2.
So the reference angle is 30°.
7 Find every angle
Sin is the up value.
The points with the height we want are on a flat line, one line for each value.
Sin is 0 on the x-axis.
x = 0° or 180°.
Sin is positive above the x-axis: quadrants I and II.
30° and 180° − 30°: x = 30° or 150°.
On the circle: 4 points, so 4 angles.
So x = 0°, 30°, 150°, 180°.
8 Check
Let's check every answer in the equation with 0 on one side.
Each one must give 0.
Every answer works.
So x = 0°, 30°, 150°, 180°.
Let's look back.
First: get 0 on one side.
Then: 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.