Exact value of tan 180°

Step by step

The question

Question: find the exact value of tan 180°.

1 Draw the unit circle

The unit circle has its center at 0 and radius 1.

We turn 180° from the positive x-axis, against the clock.

The ray meets the circle at the point P.

Here is the key fact.

P is the point (cos 180°, sin 180°).

Across is cos.

Up is sin.

2 Read the point

P is on the x-axis, one step from the center: left.

So P = (−1, 0).

Careful!

Tan is sin over cos: the up number divided by the across number.

3 Divide

Tan is sin over cos: up divided by across.

Now put the numbers in: sin 180° = 0 and cos 180° = −1.

0 divided by anything is 0.

So tan 180° = 0.

4 Check

Let's check with the calculator: tan 180°.

The calculator says 0 too.

So it is right!

Let's look back.

First: draw the unit circle.

Then: read the point.

Then: divide.

Last: check.