Calculate cos 70° cos 20° - sin 70° sin 20°

Step by step

The question

Question: find the exact value of cos 70° cos 20° − sin 70° sin 20°.

70° and 20° are not on the unit circle.

1 Spot the rule

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

With A = 70° and B = 20°.

Careful!

For cosine the sign flips.

Minus in the middle means plus between the angles.

So it is all one cos: cos(70° + 20°).

2 Add the angles

70° plus 20° is 90°.

That angle is on the unit circle!

3 Use the unit circle

From the unit circle: cos 90° = 0.

4 Check

Let's check with the calculator.

Each one rounded: the whole thing is about 0.0000.

And 0 is about 0.0000.

The same!

So it is right.

Let's look back.

First: spot the rule.

Then: add the angles.

Then: use the unit circle.

Last: check.