Exact value of tan 300° Step by Step

Step by step

The question

Question: find the exact value of tan 300°.

1 Find the quadrant

We turn 300° from the positive x-axis, against the clock.

The ray meets the circle at P.

300° is between 270° and 360°.

So it is in quadrant IV, the bottom right.

300° is in quadrant IV

2 Find the reference angle

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

Here it is 360° − 300°, which is 60°.

360° − 300° = 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 IV, P is right of the y-axis: x is positive.

P is below the x-axis: y is negative.

quadrant IV : x +, y −

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

So tan 300° = −tan 60°: the same size as for 60°, with a minus.

tan 300° = −tan 60°

4 Use the reference angle's value

From the unit circle: tan 60° = √3.

tan 60° = √3

So tan 300° = −√3.

tan 300° = −√3

5 Check

Let's check with the calculator: tan 300° is about −1.7321.

tan 300° ≈ −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