Solve sin 3x = 1/2

Step by step

The question

Question: solve sin 3x = 1/2.

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 3x, not x.

We give it a name: u = 3x.

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 3 on every part: u = 3x goes 3 times as far, from 0° up to 1080°.

1080° is 3 full turns of the circle.

So u can land on each point 3 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°.

390° + 360° is 750°.

Careful!

Do not stop after one turn!

u goes round 3 times, so every point gives 3 angles.

The same for 150°: 510° and 870°.

5 Divide by 3

u is 3x.

So x is u divided by 3.

30° ÷ 3 is 10°.

150° ÷ 3 is 50°.

And so on.

All are from 0° to 360°.

So x = 10°, 50°, 130°, 170°, 250°, 290°.

6 Check

Let's check every answer.

Put it in, times 3 first, then the unit circle.

Each one must give 1/2.

Every answer works.

So x = 10°, 50°, 130°, 170°, 250°, 290°.

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 3.

Last: check.