Exact value of sin 210° Step by Step

Step by step

The question

Question: find the exact value of sin 210°.

1 Find the quadrant

We turn 210° from the positive x-axis, against the clock.

The ray meets the circle at P.

210° is between 180° and 270°.

So it is in quadrant III, the bottom left.

210° is in quadrant III

2 Find the reference angle

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

Here it is 210° − 180°, which is 30°.

210° − 180° = 30°

Careful!

Always measure to the x-axis, never to the y-axis.

The y-axis gives 60°: the wrong triangle.

Careful: not 270° − 210° = 60° ✗

3 Find the sign

In quadrant III, P is left of the y-axis: x is negative.

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

quadrant III : x −, y −

Sin is the up number y.

In quadrant III, y is negative.

So sin 210° = −sin 30°: the same size as for 30°, with a minus.

sin 210° = −sin 30°

4 Use the reference angle's value

From the unit circle: sin 30° = 1/2.

sin 30° = 12

So sin 210° = −1/2.

sin 210° = −12

5 Check

Let's check with the calculator: sin 210° is about −0.5000.

sin 210° ≈ −0.5000

And −1/2 is about −0.5000.

The same!

So it is right.

−12 ≈ −0.5000

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