Step by step
The question
Question: solve sin 2x = 1/2.
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 sin u = 1/2: 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: sin 30° = 1/2.
So the reference angle is 30°.
Sin is the up value.
The points with that height are on a flat line.
Sin is positive above the x-axis: quadrants I and II.
30° and 180° − 30°: u = 30° or 150°.
On the circle: 2 points.
4 Add full turns
The next turn lands on the same points again: add 360°.
30° + 360° is 390°.
Careful!
Do not stop after one turn!
u goes round twice, so every point gives 2 angles.
The same for 150°: 510°.
5 Divide by 2
u is 2x.
So x is u divided by 2.
30° ÷ 2 is 15°.
150° ÷ 2 is 75°.
And so on.
All are from 0° to 360°.
6 In radians
In radians: each angle times π over 180.
15° is π/12.
75° is 5π/12.
And so on.
So x = π/12, 5π/12, 13π/12, 17π/12.
7 Check
Let's check every answer.
Put it in, times 2 first, then the unit circle.
Each one must give 1/2.
Every answer works.
So x = π/12, 5π/12, 13π/12, 17π/12.
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.