Solve 2 sin x + √3 = 0

Step by step

The question

Question: solve 2 sin x + √3 = 0.

Find every angle from 0° up to 360°.

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

1 Remove the + √3

To remove + √3, we take away √3.

We do it on both sides, so they stay equal.

+ √3 and − √3 make zero.

Cross them out.

Careful!

The √3 does not just jump across.

We took it away on both sides, so on the right it is − √3.

So 2 sin x = −√3.

A root is a number like any other.

2 Remove the 2

2 sin x means 2 times sin x.

We want just sin x.

To remove times 2, we divide by 2.

We do it on both sides, so they stay equal.

2 divided by 2 is 1.

So just sin x is left.

So sin x = −√3/2.

3 Find the reference angle

From the unit circle: sin 60° = √3/2.

So the reference angle is 60°.

The minus only changes which quadrants.

4 Find every angle

Sin is the up value.

The points with the height we want are on a flat line.

Sin is negative below the x-axis: quadrants III and IV.

180° + 60° and 360° − 60°: x = 240° or 300°.

On the circle: 2 points, so 2 angles.

So x = 240°, 300°.

5 Check

Let's check every answer in the first equation.

Each one must give 0.

Every answer works.

So x = 240°, 300°.

Let's look back.

First: remove the + √3.

Then: remove the 2.

Then: find the reference angle.

Then: find every angle.

Last: check.