Calculate arccos(-√3/2)

Step by step

The question

Question: find cos⁻¹(−√3/2).

1 What it asks

cos⁻¹ of −√3/2 means: the angle whose cosine is −√3/2.

We call that angle θ.

So cos θ = −√3/2.

2 The allowed range

Many angles have a cosine of −√3/2.

Going round, there are many.

The cos⁻¹ button must give just one.

So cos⁻¹ only gives angles from 0° to 180°: the top half of the circle.

There, every cosine from −1 to 1 happens just once.

3 Find the angle

From the unit circle: cos 30° = √3/2.

Without the minus, the angle is 30°.

Careful!

cos⁻¹ never gives a negative angle: its range is the top half.

And cos −30° is plus √3/2, not minus.

Cos is the across value.

Minus means left of the y-axis.

In the top half, that is the top left: 180° − 30°.

So θ = 150°.

4 In radians

In radians: 150 times π over 180.

That is 5π/6.

So cos⁻¹(−√3/2) = 150° = 5π/6.

5 Check

Let's check: cos 150° is −√3/2.

And 150° is inside the range.

So it is right!

Let's look back.

First: what it asks.

Then: the allowed range.

Then: find the angle.

Then: in radians.

Last: check.