Calculate cos 80° cos 20° + sin 80° sin 20°

Step by step

The question

Question: find the exact value of cos 80° cos 20° + sin 80° sin 20°.

80° 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 = 80° and B = 20°.

Careful!

For cosine the sign flips.

Plus in the middle means minus between the angles.

So it is all one cos: cos(80° − 20°).

2 Take away the angles

80° minus 20° is 60°.

That angle is on the unit circle!

3 Use the unit circle

From the unit circle: cos 60° = 1/2.

4 Check

Let's check with the calculator.

Each one rounded: the whole thing is about 0.5000.

And 1/2 is about 0.5000.

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.