Exact value of tan 120° Step by Step

Step by step

The question

Question: find the exact value of tan 120°.

1 Find the quadrant

We turn 120° from the positive x-axis, against the clock.

The ray meets the circle at P.

120° is between 90° and 180°.

So it is in quadrant II, the top left.

120° is in quadrant II

2 Find the reference angle

The reference angle is the angle between the ray and the x-axis.

Here it is 180° − 120°, which is 60°.

180° − 120° = 60°

Careful!

The reference angle is always measured to the x-axis, never to the y-axis.

Careful: measure to the x-axis

3 Find the sign

In quadrant II, P is left of the y-axis: x is negative.

P is above the x-axis: y is positive.

quadrant II : x −, y +

Tan is y divided by x: plus over minus is minus.

So tan 120° = −tan 60°: the same size as for 60°, with a minus.

tan 120° = −tan 60°

4 Use the reference angle's value

From the unit circle: tan 60° = √3.

tan 60° = √3

So tan 120° = −√3.

tan 120° = −√3

5 Check

Let's check with the calculator: tan 120° is about −1.7321.

tan 120° ≈ −1.7321

And −√3 is about −1.7321.

The same!

So it is right.

−√3 ≈ −1.7321

Let's look back.

First: find the quadrant.

Then: find the reference angle.

Then: find the sign.

Then: use the reference angle's value.

Last: check.

Answer