Step by step
The question
Question: find the exact value of sin 135°.
There are four choices.
First we solve it, the normal way.
Then we pick.
1 Find the quadrant
We turn 135° from the positive x-axis, against the clock.
The ray meets the circle at P.
The axes cut the circle into four quadrants, numbered I to IV against the clock, from the top right.
135° is between 90° and 180°.
So it is in quadrant II, the top left.
2 Find the reference angle
We know the values for 30°, 45° and 60° from the unit circle.
So we look for an angle like those.
The reference angle is the angle between the ray and the x-axis.
We measure to the nearest part of the x-axis: the left side, a half turn, 180°.
Here it is 180° − 135°, which is 45°.
The triangle from P to the x-axis has this 45° angle at the center.
It is the 45° triangle, flipped over.
Same triangle, same side lengths.
Only the signs can change.
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.
Sin is the up number y.
In quadrant II, y is positive.
So sin 135° = sin 45°: the same size as for 45°, and plus.
4 Use the reference angle's value
From the unit circle: at 45°, the point P is (√2/2, √2/2).
Sin is the up number.
So sin 45° = √2/2.
So sin 135° = √2/2.
5 Check
Let's check with the calculator: sin 135° is about 0.7071.
And √2/2 is about 0.7071.
The same!
So it is right.
6 Pick the choice
Our answer is sin 135° = √2/2.
That is choice C.
7 Why not the others
A is −√2/2.
Wrong: sin is positive between 90° and 180°.
B is 1/2.
Wrong: that is sin 30°: 135° makes 45° with the x-axis.
D is √3/2.
Wrong: that is sin 60°: 135° makes 45° with the x-axis.
Let's look back.
First: find the quadrant.
Then: find the reference angle.
Then: find the sign.
Then: use the reference angle's value.
Then: check.
Then: pick the choice.
Last: why not the others.