Exact value of sin 5π/4 Step by Step

Step by step

The question

Question: find the exact value of sin 5π/4.

1 Change to degrees

The angle is in radians.

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

5π4 = 5 × 180°4

180 divided by 4 is 45.

Times 5 is 225.

180 ÷ 4 = 45

5 × 45 = 225

So 5π/4 = 225°.

5π4 = 225°

2 Find the quadrant

We turn 225° from the positive x-axis, against the clock.

The ray meets the circle at P.

225° is between 180° and 270°.

So it is in quadrant III, the bottom left.

225° is in quadrant III

3 Find the reference angle

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

Here it is 225° − 180°, which is 45°.

225° − 180° = 45°

Careful!

The reference angle is always measured to the x-axis, never to the y-axis.

Careful: measure to the x-axis

4 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 225° = −sin 45°: the same size as for 45°, with a minus.

sin 225° = −sin 45°

5 Use the reference angle's value

From the unit circle: sin 45° = √2/2.

sin 45° = √22

So sin 5π/4 = −√2/2.

sin 5π4 = −√22

6 Check

Let's check with the calculator: sin 225° is about −0.7071.

sin 225° ≈ −0.7071

And −√2/2 is about −0.7071.

The same!

So it is right.

−√22 ≈ −0.7071

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