Step by step
The question
Question: find the exact value of cos 405°.
1 Bring it into 0° to 360°
A full turn is 360°.
Taking away a full turn lands at the same place: 45°.
Same place on the circle, same point.
So cos 405° = cos 45°.
2 Find the quadrant
We turn 45° from the positive x-axis, against the clock.
The ray meets the circle at P.
45° is between 0° and 90°.
So it is in quadrant I, the top right.
3 Find the reference angle
The reference angle is the angle between the ray and the x-axis.
Here it is 360° − 45°, which is 45°.
Careful!
The reference angle is always measured to the x-axis, never to the y-axis.
4 Find the sign
In quadrant I, P is right of the y-axis: x is positive.
P is above the x-axis: y is positive.
Cos is the across number x.
In quadrant I, x is positive.
So cos 45° = cos 45°: the same size as for 45°, and plus.
5 Use the reference angle's value
From the unit circle: cos 45° = √2/2.
So cos 405° = √2/2.
6 Check
Let's check with the calculator: cos 405° is about 0.7071.
And √2/2 is about 0.7071.
The same!
So it is right.
Let's look back.
First: bring it into 0° to 360°.
Then: find the quadrant.
Then: find the reference angle.
Then: find the sign.
Then: use the reference angle's value.
Last: check.