Step by step
The question
Question: solve cos 2x = −1/2.
Find every angle from 0 up to 2π.
From 0 to 2π: one full turn.
We work in degrees, 0° to 360°, and change to radians at the end.
1 Name the angle
The angle here is 2x, not x.
We give it a name: u = 2x.
Then the equation is cos u = −1/2: 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 Find u in the first turn
From the unit circle: cos 60° = 1/2.
So the reference angle is 60°.
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° − 60° and 180° + 60°: u = 120° or 240°.
On the circle: 2 points.
4 Add full turns
The next turn lands on the same points again: add 360°.
120° + 360° is 480°.
Careful!
Do not stop after one turn!
u goes round twice, so every point gives 2 angles.
The same for 240°: 600°.
5 Divide by 2
u is 2x.
So x is u divided by 2.
120° ÷ 2 is 60°.
240° ÷ 2 is 120°.
And so on.
All are from 0° to 360°.
6 In radians
In radians: each angle times π over 180.
60° is π/3.
120° is 2π/3.
And so on.
So x = π/3, 2π/3, 4π/3, 5π/3.
7 Check
Let's check every answer.
Put it in, times 2 first, then the unit circle.
Each one must give −1/2.
Every answer works.
So x = π/3, 2π/3, 4π/3, 5π/3.
Let's look back.
First: name the angle.
Then: find how far u goes.
Then: find u in the first turn.
Then: add full turns.
Then: divide by 2.
Then: in radians.
Last: check.