Exact value of cos 225° Step by Step

Step by step

The question

Question: find the exact value of cos 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 −

Cos is the across number x.

In quadrant III, x is negative.

So cos 225° = −cos 45°: the same size as for 45°, with a minus.

cos 225° = −cos 45°

4 Use the reference angle's value

From the unit circle: cos 45° = √2/2.

cos 45° = √22

So cos 225° = −√2/2.

cos 225° = −√22

5 Check

Let's check with the calculator: cos 225° is about −0.7071.

cos 225° ≈ −0.7071

And −√2/2 is about −0.7071.

The same!

So it is right.

−√22 ≈ −0.7071

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