Calculate cos^-1(-1/2)

Step by step

The question

Question: find cos⁻¹(−1/2).

1 What it asks

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

We call that angle θ.

So cos θ = −1/2.

2 The allowed range

Many angles have a cosine of −1/2.

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 60° = 1/2.

Without the minus, the angle is 60°.

Careful!

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

And cos −60° is plus 1/2, not minus.

Cos is the across value.

Minus means left of the y-axis.

In the top half, that is the top left: 180° − 60°.

So θ = 120°.

4 In radians

In radians: 120 times π over 180.

That is 2π/3.

So cos⁻¹(−1/2) = 120° = 2π/3.

5 Check

Let's check: cos 120° is −1/2.

And 120° 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.