Calculate tan^-1(1)

Step by step

The question

Question: find tan⁻¹(1).

1 What it asks

tan⁻¹ of 1 means: the angle whose tangent is 1.

We call that angle θ.

So tan θ = 1.

2 The allowed range

Many angles have a tangent of 1.

225° and 405° work too.

The tan⁻¹ button must give just one.

So tan⁻¹ gives an angle between −90° and 90°: the right half of the circle.

Not −90° or 90° themselves: tan has no value there.

3 Find the angle

From the unit circle: tan 45° = 1.

Careful!

Other angles have the same tan, like 225°.

But only 45° is in the range.

So the button gives 45°.

45° is in the range.

So θ = 45°.

4 In radians

In radians: 45 times π over 180.

That is π/4.

So tan⁻¹(1) = 45° = π/4.

5 Check

Let's check: tan 45° is 1.

And 45° 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.