Solve sin 3x = 1 Step by Step | Multiple-angle equations

Step by step

The question

Question: solve sin 3x = 1.

Find every angle from 0° up to 360°.

sin 3x = 1

We want the angles from 0° to 360°: one full turn.

0° ≤ x < 360°

1 Name the angle

The angle here is 3x, not x.

We give it a name: u = 3x.

u = 3x

Then the equation is sin u = 1: the kind we can already solve.

sin u = 1

2 Find how far u goes

x goes from 0° to 360°.

0° ≤ x < 360°

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

0° ≤ 3x < 1080°

1080° is 3 full turns of the circle.

So u can land on each point 3 times.

1080° = 3 full turns

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°.

sin u = 1: 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°.

450° + 360° is 810°.

u = 90°, 450°, 810°

Careful!

Do not stop after one turn!

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

Careful: not only u = 90° ✗

5 Divide by 3

u is 3x.

So x is u divided by 3.

x = u ÷ 3

90° ÷ 3 is 30°.

450° ÷ 3 is 150°.

And so on.

All are from 0° to 360°.

So x = 30°, 150°, 270°.

x = 30°, 150°, 270°

6 Check

Let's check every answer.

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

Each one must give 1.

x = 30°: sin(3 × 30°) = sin 90° = 1

x = 150°: sin(3 × 150°) = sin 450° = 1

x = 270°: sin(3 × 270°) = sin 810° = 1

Every answer works.

So x = 30°, 150°, 270°.

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.

Answer