Exact value of tan 225° Step by Step

Step by step

The question

Question: find the exact value of tan 225°.

1 Find the quadrant

We turn 225° from the positive x-axis, against the clock.

The ray meets the circle at P.

225° is between 180° and 270°.

So it is in quadrant III, the bottom left.

225° is in quadrant III

2 Find the reference angle

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

Here it is 225° − 180°, which is 45°.

225° − 180° = 45°

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 III, P is left of the y-axis: x is negative.

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

quadrant III : x −, y −

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

So tan 225° = tan 45°: the same size as for 45°, and plus.

tan 225° = tan 45°

4 Use the reference angle's value

From the unit circle: tan 45° = 1.

tan 45° = 1

So tan 225° = 1.

tan 225° = 1

5 Check

Let's check with the calculator: tan 225° is about 1.0000.

tan 225° ≈ 1.0000

And 1 is about 1.0000.

The same!

So it is right.

1 ≈ 1.0000

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