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.

300° and 420° work too.

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.

Careful!

Other angles have the same cos, like 300°.

But only 60° is in the range.

So the button gives 60°.

60° is in the range.

So θ = 60°.

4 In radians

In radians: 60 times π over 180.

That is π/3.

So cos⁻¹(1/2) = 60° = π/3.

5 Check

Let's check: cos 60° is 1/2.

And 60° 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.