Solve tan 2x = 1

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°.

Tan is positive in quadrants I and III.

45° and 180° + 45°: u = 45° or 225°.

On the circle: 2 points.

4 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 225°: 585°.

5 Divide by 2

u is 2x.

So x is u divided by 2.

45° ÷ 2 is 22.5°.

225° ÷ 2 is 112.5°.

And so on.

All are from 0° to 360°.

So x = 22.5°, 112.5°, 202.5°, 292.5°.

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 = 22.5°, 112.5°, 202.5°, 292.5°.

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.