Step by step
The question
Question: solve tan 2x = 1.
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 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°.
6 In radians
In radians: each angle times π over 180.
22.5° is 23π/180.
112.5° is 113π/180.
And so on.
So x = 23π/180, 113π/180, 203π/180, 293π/180.
7 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 = 23π/180, 113π/180, 203π/180, 293π/180.
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.