Calculate tan^-1(-1) Step by Step

Step by step

The question

Question: find tan⁻¹(−1).

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.

tan θ = −1

2 The allowed range

Many angles have a tangent of −1.

Going round, there are many.

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.

−90° < θ < 90°

3 Find the angle

From the unit circle: tan 45° = 1.

Without the minus, the angle is 45°.

tan 45° = 1

Careful!

315° has the right tan, but it is not between −90° and 90°.

Below the x-axis on the right, inside the range, is −45°.

Careful: not θ = 315° ✗

Tan is up over across.

Minus means below the x-axis on the right.

In the right half, that is turning 45° the other way: θ = −45°.

θ = −45°

4 In radians

In radians: −45 times π over 180.

That is −π/4.

θ = −45 × π180 = −π4

So tan⁻¹(−1) = −45° = −π/4.

tan⁻¹(−1) = −45° = −π4

5 Check

Let's check: tan −45° is −1.

tan −45° = −1

And −45° is inside the range.

So it is right!

−90° < −45° < 90°

Let's look back.

First: what it asks.

Then: the allowed range.

Then: find the angle.

Then: in radians.

Last: check.

Answer