Exact value of sin 315° Step by Step

Step by step

The question

Question: find the exact value of sin 315°.

1 Find the quadrant

We turn 315° from the positive x-axis, against the clock.

The ray meets the circle at P.

315° is between 270° and 360°.

So it is in quadrant IV, the bottom right.

315° is in quadrant IV

2 Find the reference angle

The reference angle is the angle between the ray and the x-axis.

Here it is 360° − 315°, which is 45°.

360° − 315° = 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 IV, P is right of the y-axis: x is positive.

P is below the x-axis: y is negative.

quadrant IV : x +, y −

Sin is the up number y.

In quadrant IV, y is negative.

So sin 315° = −sin 45°: the same size as for 45°, with a minus.

sin 315° = −sin 45°

4 Use the reference angle's value

From the unit circle: sin 45° = √2/2.

sin 45° = √22

So sin 315° = −√2/2.

sin 315° = −√22

5 Check

Let's check with the calculator: sin 315° is about −0.7071.

sin 315° ≈ −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