Exact value of cos 165° Step by Step | Sum and difference identities

Step by step

The question

Question: find the exact value of cos 165°.

Exact means with roots and fractions, not a decimal.

cos 165° =

1 Split the angle

165° is not on the unit circle.

But 120° and 45° are, and 120° plus 45° is 165°.

165° = 120° + 45°

So cos 165° is cos of 120° plus 45°.

= cos(120° + 45°)

2 Write the rule

The sum rule for cosine: cos(A + B) = cos A cos B − sin A sin B.

Look at the sign in the middle: it is the other one, minus.

cos(A + B) = cos A cos B − sin A sin B

Careful!

For cosine the sign in the middle flips.

A plus inside gives minus in the middle.

Careful: cos(A + B): − in the middle

A is 120° and B is 45°.

We put them in the rule.

= cos 120° cos 45° − sin 120° sin 45°

3 Put in the values

From the unit circle: cos 120° = −1/2.

cos 45° = √2/2.

sin 120° = √3/2.

sin 45° = √2/2.

120° is in quadrant II: the signs come from there.

cos 120° = −12, sin 120° = √32

cos 45° = √22, sin 45° = √22

We put the values in.

= −12 × √22 − √32 × √22

4 Multiply

Tops times tops, bottoms times bottoms.

A root times a root: √3 × √2 = √6.

√3 × √2 = √6

−1/2 times √2/2 is −√2/4.

√3/2 times √2/2 is √6/4.

= −√24 − √64

5 Make one fraction

Both have the same bottom, 4.

So we put the tops together over 4: (−√6 − √2)/4.

= −√6 − √24

So cos 165° = (−√6 − √2)/4.

cos 165° = −√6 − √24

6 Check

Let's check with the calculator.

Our answer: √6 is about 2.4495, √2 is about 1.4142.

Over 4: about −0.9659.

−√6 − √24 ≈ (−2.4495 − 1.4142) ÷ 4 ≈ −0.9659

And cos 165° on the calculator is about −0.9659.

The same!

So it is right.

cos 165° ≈ −0.9659

Let's look back.

First: split the angle.

Then: write the rule.

Then: put in the values.

Then: multiply.

Then: make one fraction.

Last: check.

Answer