Exact value of cos 345° Step by Step | Sum and difference identities

Step by step

The question

Question: find the exact value of cos 345°.

Exact means with roots and fractions, not a decimal.

cos 345° =

1 Split the angle

345° is not on the unit circle.

But 300° and 45° are, and 300° plus 45° is 345°.

345° = 300° + 45°

So cos 345° is cos of 300° plus 45°.

= cos(300° + 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 300° and B is 45°.

We put them in the rule.

= cos 300° cos 45° − sin 300° sin 45°

3 Put in the values

From the unit circle: cos 300° = 1/2.

cos 45° = √2/2.

sin 300° = −√3/2.

sin 45° = √2/2.

300° is in quadrant IV: the signs come from there.

cos 300° = 12, sin 300° = −√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.

Taking away a negative is adding.

= √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 345° = (√6 + √2)/4.

cos 345° = √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 345° on the calculator is about 0.9659.

The same!

So it is right.

cos 345° ≈ 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