Exact value of cos 11π/12 Step by Step | Sum and difference identities

Step by step

The question

Question: find the exact value of cos 11π/12.

Exact means with roots and fractions, not a decimal.

cos 11π12 =

1 Split the angle

11π/12 is not on the unit circle.

But 2π/3 and π/4 are, and 2π/3 plus π/4 is 11π/12.

11π12 = 8π12 + 3π12 = 2π3 + π4

So cos 11π/12 is cos of 2π/3 plus π/4.

= cos(2π3 + π4)

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 2π/3 and B is π/4.

We put them in the rule.

= cos 2π3 cos π4 − sin 2π3 sin π4

3 Put in the values

From the unit circle: cos 2π/3 = −1/2.

cos π/4 = √2/2.

sin 2π/3 = √3/2.

sin π/4 = √2/2.

2π/3 is in quadrant II: the signs come from there.

cos 2π3 = −12, sin 2π3 = √32

cos π4 = √22, sin π4 = √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 11π/12 = (−√6 − √2)/4.

cos 11π12 = −√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 11π/12 on the calculator is about −0.9659.

The same!

So it is right.

cos 11π12 ≈ −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