Step by step
The question
Question: solve 2 cos 2x = √2.
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 = √2: 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 = √2/2.
4 Find u in the first turn
From the unit circle: cos 45° = √2/2.
So the reference angle is 45°.
Cos is the across value.
The points with that across value are on an up-and-down line.
Cos is positive right of the y-axis: quadrants I and IV.
45° and 360° − 45°: u = 45° or 315°.
On the circle: 2 points.
5 Add full turns
The next turn lands on the same points again: add 360°.
45° + 360° is 405°.
Careful!
Do not stop after one turn!
u goes round twice, so every point gives 2 angles.
The same for 315°: 675°.
6 Divide by 2
u is 2x.
So x is u divided by 2.
45° ÷ 2 is 22.5°.
315° ÷ 2 is 157.5°.
And so on.
All are from 0° to 360°.
So x = 22.5°, 157.5°, 202.5°, 337.5°.
7 Check
Let's check every answer.
Put it in, times 2 first, then the unit circle.
Each one must give √2.
Every answer works.
So x = 22.5°, 157.5°, 202.5°, 337.5°.
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.