Exact value of sin 120° Step by Step

Step by step

The question

Question: find the exact value of sin 120°.

1 Find the quadrant

We turn 120° from the positive x-axis, against the clock.

The ray meets the circle at P.

120° is between 90° and 180°.

So it is in quadrant II, the top left.

120° is in quadrant II

2 Find the reference angle

The reference angle is the angle between the ray and the x-axis.

Here it is 180° − 120°, which is 60°.

180° − 120° = 60°

Careful!

Always measure to the x-axis, never to the y-axis.

The y-axis gives 30°: the wrong triangle.

Careful: not 120° − 90° = 30° ✗

3 Find the sign

In quadrant II, P is left of the y-axis: x is negative.

P is above the x-axis: y is positive.

quadrant II : x −, y +

Sin is the up number y.

In quadrant II, y is positive.

So sin 120° = sin 60°: the same size as for 60°, and plus.

sin 120° = sin 60°

4 Use the reference angle's value

From the unit circle: sin 60° = √3/2.

sin 60° = √32

So sin 120° = √3/2.

sin 120° = √32

5 Check

Let's check with the calculator: sin 120° is about 0.8660.

sin 120° ≈ 0.8660

And √3/2 is about 0.8660.

The same!

So it is right.

√32 ≈ 0.8660

Let's look back.

First: find the quadrant.

Then: find the reference angle.

Then: find the sign.

Then: use the reference angle's value.

Last: check.

Answer