Step by step
The question
Question: solve 2 sin² x = 1.
Find every angle from 0° up to 360°.
We want the angles from 0° to 360°: one full turn.
1 Remove the 2
2 sin² x means 2 times sin² x.
We want just sin² x.
To remove times 2, we divide by 2.
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.
2 divided by 2 is 1.
So just sin² x is left.
So sin² x = 1/2!
2 Square root of both sides
sin x squared is 1/2.
So sin x is the square root of 1/2, or minus it.
Both work.
The root of a fraction: root of the top over root of the bottom.
1 over √2: times √2 over √2 is √2/2.
Careful!
A square root gives two answers: plus and minus.
sin x = −√2/2 works too.
So sin x = √2/2 or sin x = −√2/2.
3 Find the reference angle
From the unit circle: sin 45° = √2/2.
So the reference angle is 45°.
The minus only changes which quadrants.
4 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 positive above the x-axis: quadrants I and II.
45° and 180° − 45°: x = 45° or 135°.
Sin is negative below the x-axis: quadrants III and IV.
180° + 45° and 360° − 45°: x = 225° or 315°.
On the circle: 4 points, so 4 angles.
So x = 45°, 135°, 225°, 315°.
5 Check
Let's check every answer in the equation.
Each one must give 1.
Every answer works.
So x = 45°, 135°, 225°, 315°.
Let's look back.
First: remove the 2.
Then: square root of both sides.
Then: find the reference angle.
Then: find every angle.
Last: check.