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

Step by step

The question

Question: find the exact value of sin π/12.

Exact means with roots and fractions, not a decimal.

sin π12 =

1 Split the angle

π/12 is not on the unit circle.

But π/4 and π/6 are, and π/4 minus π/6 is π/12.

π12 = 3π12 − 2π12 = π4 − π6

So sin π/12 is sin of π/4 minus π/6.

= sin(π4 − π6)

2 Write the rule

The difference rule for sine: sin(A − B) = sin A cos B − cos A sin B.

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

Careful!

We cannot split sin into two sins.

We use the rule.

Careful: not sin π12 = sin π4 − sin π6 ✗

A is π/4 and B is π/6.

We put them in the rule.

= sin π4 cos π6 − cos π4 sin π6

3 Put in the values

From the unit circle: sin π/4 = √2/2.

cos π/6 = √3/2.

cos π/4 = √2/2.

sin π/6 = 1/2.

sin π4 = √22, cos π4 = √22

cos π6 = √32, sin π6 = 12

We put the values in.

= √22 × √32 − √22 × 12

4 Multiply

Tops times tops, bottoms times bottoms.

A root times a root: √2 × √3 = √6.

√2 × √3 = √6

√2/2 times √3/2 is √6/4.

√2/2 times 1/2 is √2/4.

= √64 − √24

5 Make one fraction

Both have the same bottom, 4.

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

= √6 − √24

So sin π/12 = (√6 − √2)/4.

sin π12 = √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 0.2588.

√6 − √24 ≈ (2.4495 − 1.4142) ÷ 4 ≈ 0.2588

And sin π/12 on the calculator is about 0.2588.

The same!

So it is right.

sin π12 ≈ 0.2588

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