Step by step
The question
Question: solve 2 sin x + 1 = 0.
Find every angle from 0° up to 360°.
We want the angles from 0° to 360°: one full turn.
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.
1 Remove the + 1
To remove + 1, we take away 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.
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 = −1/2!
3 Find the reference angle
From the unit circle: sin 30° = 1/2.
So the reference angle is 30°.
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.
Sin is negative below the x-axis: quadrants III and IV.
180° + 30° and 360° − 30°: x = 210° or 330°.
Careful!
The calculator gives only 210°.
But the line crosses the circle twice.
330° works too.
On the circle: 2 points, so 2 angles.
So x = 210°, 330°.
5 Check
Let's check every answer in the equation.
Each one must give 0.
Every answer works.
So x = 210°, 330°.
Let's look back.
First: remove the + 1.
Then: remove the 2.
Then: find the reference angle.
Then: find every angle.
Last: check.