Step by step
The question
Question: solve tan 2x = −1.
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 tan u = −1: 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: tan 45° = 1.
So the reference angle is 45°.
The minus only changes which quadrants.
Tan is negative in quadrants II and IV.
180° − 45° and 360° − 45°: u = 135° or 315°.
On the circle: 2 points.
4 Add full turns
The next turn lands on the same points again: add 360°.
135° + 360° is 495°.
Careful!
Do not stop after one turn!
u goes round twice, so every point gives 2 angles.
The same for 315°: 675°.
5 Divide by 2
u is 2x.
So x is u divided by 2.
135° ÷ 2 is 67.5°.
315° ÷ 2 is 157.5°.
And so on.
All are from 0° to 360°.
So x = 67.5°, 157.5°, 247.5°, 337.5°.
6 Check
Let's check every answer.
Put it in, times 2 first, then the unit circle.
Each one must give −1.
Every answer works.
So x = 67.5°, 157.5°, 247.5°, 337.5°.
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.
Last: check.