Exact value of sin 135° Step by Step

Step by step

The question

Question: find the exact value of sin 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.

135° is in quadrant II

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°.

180° − 135° = 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 II, P is left of the y-axis: x is negative.

P is above the x-axis: y is positive.

quadrant II : x −, y +

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.

sin 135° = sin 45°

4 Use the reference angle's value

From the unit circle: sin 45° = √2/2.

sin 45° = √22

So sin 135° = √2/2.

sin 135° = √22

5 Check

Let's check with the calculator: sin 135° is about 0.7071.

sin 135° ≈ 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