Calculate sin 140° cos 20° - cos 140° sin 20°

Step by step

The question

Question: find the exact value of sin 140° cos 20° − cos 140° sin 20°.

140° and 20° are not on the unit circle.

1 Spot the rule

This is the right side of a rule we know: sin(A − B) = sin A cos B − cos A sin B.

With A = 140° and B = 20°.

Careful!

The angle in the first sin is A.

So A is 140° and B is 20°.

The order matters when we take away.

So it is all one sin: sin(140° − 20°).

2 Take away the angles

140° minus 20° is 120°.

That angle is on the unit circle!

3 Use the unit circle

From the unit circle: sin 120° = √3/2.

4 Check

Let's check with the calculator.

Each one rounded: the whole thing is about 0.8660.

And √3/2 is about 0.8660.

The same!

So it is right.

Let's look back.

First: spot the rule.

Then: take away the angles.

Then: use the unit circle.

Last: check.