The question
Question: solve 2 sin x − 1 = 0.
Find every angle from 0° up to 360°.
2 sin x − 1 = 0
We want the angles from 0° to 360°: one full turn.
0° ≤ x < 360°
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 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.
2 sin x − 1 + 1 = 0 + 1
− 1 and + 1 make zero.
Cross them out.
0 + 1 = 1.
0 + 1 = 1
Now we have 2 sin x = 1.
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 sin x2 = 12
2 divided by 2 is 1.
So just sin x is left.
So sin x = 1/2!
sin x = 12
3 Find the reference angle
From the unit circle: sin 30° = 1/2.
So the reference angle is 30°.
sin 30° = 12
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.
30° and 180° − 30°: x = 30° or 150°.
sin x = 12: x = 30° or 150°
Careful!
The calculator gives only 30°.
But the line crosses the circle twice.
150° works too.
Careful: not only x = 30° ✗
On the circle: 2 points, so 2 angles.
So x = 30°, 150°.
x = 30°, 150°
5 Check
Let's check every answer in the equation.
Each one must give 0.
x = 30°: 2 sin 30° − 1 = 2 × 12 − 1 = 0
x = 150°: 2 sin 150° − 1 = 2 × 12 − 1 = 0
Every answer works.
So x = 30°, 150°.
Let's look back.
First: remove the − 1.
Then: remove the 2.
Then: find the reference angle.
Then: find every angle.
Last: check.
▶ Watch it solved
The question
Question: solve 2 cos x = √3.
Find every angle from 0° up to 360°.
2 cos x = √3
We want the angles from 0° to 360°: one full turn.
0° ≤ x < 360°
1 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.
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 cos x2 = √32
2 divided by 2 is 1.
So just cos x is left.
So cos x = √3/2.
cos x = √32
2 Find the reference angle
From the unit circle: cos 30° = √3/2.
So the reference angle is 30°.
cos 30° = √32
3 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.
30° and 360° − 30°: x = 30° or 330°.
cos x = √32: x = 30° or 330°
On the circle: 2 points, so 2 angles.
So x = 30°, 330°.
x = 30°, 330°
4 Check
Let's check every answer in the equation.
Each one must give √3.
x = 30°: 2 cos 30° = 2 × √32 = √3
x = 330°: 2 cos 330° = 2 × √32 = √3
Every answer works.
So x = 30°, 330°.
Let's look back.
First: remove the 2.
Then: find the reference angle.
Then: find every angle.
Last: check.
▶ Watch it solved
The question
Question: solve 2 sin² x − sin x = 0.
Find every angle from 0° up to 360°.
2 sin2 x − sin x = 0
We want the angles from 0° to 360°: one full turn.
0° ≤ x < 360°
Careful!
Do not divide by sin x!
That throws away sin x = 0, and its answers.
We take sin x out instead.
Careful: not ÷ sin x ✗
1 Take out the common factor
Both parts have sin x in them.
We take it out, like a common factor.
sin x(2 sin x − 1) = 0
Check the first part: sin x times 2 sin x is 2 sin squared x.
sin x × 2 sin x = 2 sin² x
2 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...
sin x = 0
...
2 sin x − 1 = 0.
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.
3 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.
2 sin x − 1 + 1 = 0 + 1
− 1 and + 1 make zero.
Cross them out.
0 + 1 = 1.
0 + 1 = 1
Now we have 2 sin x = 1.
2 sin x = 1
4 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 sin x2 = 12
2 divided by 2 is 1.
So just sin x is left.
So sin x = 1/2!
sin x = 12
5 Find the reference angle
From the unit circle: sin 30° = 1/2.
So the reference angle is 30°.
sin 30° = 12
6 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 x = 0: x = 0° or 180°
Sin is positive above the x-axis: quadrants I and II.
30° and 180° − 30°: x = 30° or 150°.
sin x = 12: x = 30° or 150°
On the circle: 4 points, so 4 angles.
So x = 0°, 30°, 150°, 180°.
x = 0°, 30°, 150°, 180°
7 Check
Let's check every answer in the equation.
Each one must give 0.
x = 0°: 2 sin² 0° − sin 0° = 2 × (0)² − 0 = 0
x = 30°: 2 sin² 30° − sin 30° = 2 × (12)² − 12 = 0
x = 150°: 2 sin² 150° − sin 150° = 2 × (12)² − 12 = 0
x = 180°: 2 sin² 180° − sin 180° = 2 × (0)² − 0 = 0
Every answer works.
So x = 0°, 30°, 150°, 180°.
Let's look back.
First: 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.
Answer x = 0°, 30°, 150°, 180°
▶ Watch it solved
