Calculate tan^-1(-√3/3)

Step by step

The question

Question: find tan⁻¹(−√3/3).

1 What it asks

tan⁻¹ of −√3/3 means: the angle whose tangent is −√3/3.

We call that angle θ.

So tan θ = −√3/3.

2 The allowed range

Many angles have a tangent of −√3/3.

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.

3 Find the angle

From the unit circle: tan 30° = √3/3.

Without the minus, the angle is 30°.

Careful!

330° has the right tan, but it is not between −90° and 90°.

Below the x-axis on the right, inside the range, is −30°.

Tan is up over across.

Minus means below the x-axis on the right.

In the right half, that is turning 30° the other way: θ = −30°.

4 In radians

In radians: −30 times π over 180.

That is −π/6.

So tan⁻¹(−√3/3) = −30° = −π/6.

5 Check

Let's check: tan −30° is −√3/3.

And −30° 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.