Step by step
The question
Question: solve 2 cos 2x = −√3.
Find every angle from 0° up to 360°.
We want the angles from 0° to 360°: one full turn.
1 Name the angle
The angle here is 2x, not x.
We give it a name: u = 2x.
Then the equation is 2 cos u = −√3: the kind we can already solve.
2 Find how far u goes
x goes from 0° to 360°.
Times 2 on every part: u = 2x goes twice as far, from 0° up to 720°.
720° is 2 full turns of the circle.
So u can land on each point 2 times.
3 Remove the 2
2 cos u means 2 times cos u.
We want just cos u.
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 cos u is left.
So cos u = −√3/2.
4 Find u in the first turn
From the unit circle: cos 30° = √3/2.
So the reference angle is 30°.
The minus only changes which quadrants.
Cos is the across value.
The points with that across value are on an up-and-down line.
Cos is negative left of the y-axis: quadrants II and III.
180° − 30° and 180° + 30°: u = 150° or 210°.
On the circle: 2 points.
5 Add full turns
The next turn lands on the same points again: add 360°.
150° + 360° is 510°.
Careful!
Do not stop after one turn!
u goes round twice, so every point gives 2 angles.
The same for 210°: 570°.
6 Divide by 2
u is 2x.
So x is u divided by 2.
150° ÷ 2 is 75°.
210° ÷ 2 is 105°.
And so on.
All are from 0° to 360°.
So x = 75°, 105°, 255°, 285°.
7 Check
Let's check every answer.
Put it in, times 2 first, then the unit circle.
Each one must give −√3.
Every answer works.
So x = 75°, 105°, 255°, 285°.
Let's look back.
First: name the angle.
Then: find how far u goes.
Then: remove the 2.
Then: find u in the first turn.
Then: add full turns.
Then: divide by 2.
Last: check.