Exact value of sin 5π/3 Step by Step

Step by step

The question

Question: find the exact value of sin 5π/3.

1 Change to degrees

The angle is in radians.

π is 180°, so we put 180° in place of π.

5π3 = 5 × 180°3

180 divided by 3 is 60.

Times 5 is 300.

180 ÷ 3 = 60

5 × 60 = 300

So 5π/3 = 300°.

5π3 = 300°

2 Find the quadrant

We turn 300° from the positive x-axis, against the clock.

The ray meets the circle at P.

300° is between 270° and 360°.

So it is in quadrant IV, the bottom right.

300° is in quadrant IV

3 Find the reference angle

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

Here it is 360° − 300°, which is 60°.

360° − 300° = 60°

Careful!

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

The y-axis gives 30°: the wrong triangle.

Careful: not 300° − 270° = 30° ✗

4 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 300° = −sin 60°: the same size as for 60°, with a minus.

sin 300° = −sin 60°

5 Use the reference angle's value

From the unit circle: sin 60° = √3/2.

sin 60° = √32

So sin 5π/3 = −√3/2.

sin 5π3 = −√32

6 Check

Let's check with the calculator: sin 300° is about −0.8660.

sin 300° ≈ −0.8660

And −√3/2 is about −0.8660.

The same!

So it is right.

−√32 ≈ −0.8660

Let's look back.

First: change to degrees.

Then: find the quadrant.

Then: find the reference angle.

Then: find the sign.

Then: use the reference angle's value.

Last: check.

Answer