Calculate cos^-1(-1)

Step by step

The question

Question: find cos⁻¹(−1).

1 What it asks

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

We call that angle θ.

So cos θ = −1.

2 The allowed range

Many angles have a cosine of −1.

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 0° = 1.

Without the minus, the angle is 0°.

Careful!

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

Cos is the across value.

Across is −1 only at the far left of the circle: θ = 180°.

4 In radians

In radians: 180 times π over 180.

That is π.

So cos⁻¹(−1) = 180° = π.

5 Check

Let's check: cos 180° is −1.

And 180° 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.