Step by step
The question
Question: find the exact value of cos 135°.
1 Find the quadrant
We turn 135° from the positive x-axis, against the clock.
The ray meets the circle at P.
135° is between 90° and 180°.
So it is in quadrant II, the top left.
2 Find the reference angle
The reference angle is the angle between the ray and the x-axis.
Here it is 180° − 135°, which is 45°.
Careful!
The reference angle is always measured to the x-axis, never to the y-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.
Cos is the across number x.
In quadrant II, x is negative.
So cos 135° = −cos 45°: the same size as for 45°, with a minus.
4 Use the reference angle's value
From the unit circle: cos 45° = √2/2.
So cos 135° = −√2/2.
5 Check
Let's check with the calculator: cos 135° is about −0.7071.
And −√2/2 is about −0.7071.
The same!
So it is right.
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.