Step by step
The question
Question: solve sin 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 sin 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
Sin is the up value.
The points with that height are on a flat line.
The up value is 1 at one point only.
u = 90°.
On the circle: 1 point.
4 Add full turns
The next turn lands on the same points again: add 360°.
90° + 360° is 450°.
Careful!
Do not stop after one turn!
u goes round twice, so every point gives 2 angles.
5 Divide by 2
u is 2x.
So x is u divided by 2.
90° ÷ 2 is 45°.
450° ÷ 2 is 225°.
And so on.
All are from 0° to 360°.
So x = 45°, 225°.
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 = 45°, 225°.
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.