Exact value of cos 150° Step by Step

Step by step

The question

Question: find the exact value of cos 150°.

1 Find the quadrant

We turn 150° from the positive x-axis, against the clock.

The ray meets the circle at P.

150° is between 90° and 180°.

So it is in quadrant II, the top left.

150° 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° − 150°, which is 30°.

180° − 150° = 30°

Careful!

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

The y-axis gives 60°: the wrong triangle.

Careful: not 150° − 90° = 60° ✗

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 +

Cos is the across number x.

In quadrant II, x is negative.

So cos 150° = −cos 30°: the same size as for 30°, with a minus.

cos 150° = −cos 30°

4 Use the reference angle's value

From the unit circle: cos 30° = √3/2.

cos 30° = √32

So cos 150° = −√3/2.

cos 150° = −√32

5 Check

Let's check with the calculator: cos 150° is about −0.8660.

cos 150° ≈ −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