Step by step
The question
Question: solve cos 3x = −1/2.
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 3x, not x.
We give it a name: u = 3x.
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 3 on every part: u = 3x goes 3 times as far, from 0° up to 1080°.
1080° is 3 full turns of the circle.
So u can land on each point 3 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°.
480° + 360° is 840°.
Careful!
Do not stop after one turn!
u goes round 3 times, so every point gives 3 angles.
The same for 240°: 600° and 960°.
5 Divide by 3
u is 3x.
So x is u divided by 3.
120° ÷ 3 is 40°.
240° ÷ 3 is 80°.
And so on.
All are from 0° to 360°.
So x = 40°, 80°, 160°, 200°, 280°, 320°.
6 Check
Let's check every answer.
Put it in, times 3 first, then the unit circle.
Each one must give −1/2.
Every answer works.
So x = 40°, 80°, 160°, 200°, 280°, 320°.
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 3.
Last: check.