Exact value of sin 15° Step by Step | Sum and difference identities

Step by step

The question

Question: find the exact value of sin 15°.

Exact means with roots and fractions, not a decimal.

sin 15° =

1 Split the angle

15° is not on the unit circle.

But 45° and 30° are, and 45° minus 30° is 15°.

15° = 45° − 30°

So sin 15° is sin of 45° minus 30°.

= sin(45° − 30°)

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 15° = sin 45° − sin 30° ✗

A is 45° and B is 30°.

We put them in the rule.

= sin 45° cos 30° − cos 45° sin 30°

3 Put in the values

From the unit circle: sin 45° = √2/2.

cos 30° = √3/2.

cos 45° = √2/2.

sin 30° = 1/2.

sin 45° = √22, cos 45° = √22

cos 30° = √32, sin 30° = 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 15° = (√6 − √2)/4.

sin 15° = √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 15° on the calculator is about 0.2588.

The same!

So it is right.

sin 15° ≈ 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