Step by step
The question
Question: solve 2 sin x − √2 = 0.
Find every angle from 0° up to 360°.
We want the angles from 0° to 360°: one full turn.
1 Remove the − √2
To remove − √2, we add √2.
We do it on both sides, so they stay equal.
− √2 and + √2 make zero.
Cross them out.
Careful!
The √2 does not just jump across.
We added it on both sides, so on the right it is + √2.
So 2 sin x = √2.
A root is a number like any other.
2 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 = √2/2.
3 Find the reference angle
From the unit circle: sin 45° = √2/2.
So the reference angle is 45°.
4 Find every angle
Sin is the up value.
The points with the height we want are on a flat line.
Sin is positive above the x-axis: quadrants I and II.
45° and 180° − 45°: x = 45° or 135°.
On the circle: 2 points, so 2 angles.
So x = 45°, 135°.
5 Check
Let's check every answer in the first equation.
Each one must give 0.
Every answer works.
So x = 45°, 135°.
Let's look back.
First: remove the − √2.
Then: remove the 2.
Then: find the reference angle.
Then: find every angle.
Last: check.