Exact value of sin 105° Step by Step | Sum and difference identities

Step by step

The question

Question: find the exact value of sin 105°.

Exact means with roots and fractions, not a decimal.

sin 105° =

1 Split the angle

105° is not on the unit circle.

But 60° and 45° are, and 60° plus 45° is 105°.

105° = 60° + 45°

So sin 105° is sin of 60° plus 45°.

= sin(60° + 45°)

2 Write the rule

The sum 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.

sin 60° + sin 45° is about 1.5731: more than 1.

A sine is never more than 1.

We use the rule.

Careful: not sin 105° = sin 60° + sin 45° ✗

A is 60° and B is 45°.

We put them in the rule.

= sin 60° cos 45° + cos 60° sin 45°

3 Put in the values

From the unit circle: sin 60° = √3/2.

cos 45° = √2/2.

cos 60° = 1/2.

sin 45° = √2/2.

sin 60° = √32, cos 60° = 12

cos 45° = √22, sin 45° = √22

We put the values in.

= √32 × √22 + 12 × √22

4 Multiply

Tops times tops, bottoms times bottoms.

A root times a root: √3 × √2 = √6.

√3 × √2 = √6

√3/2 times √2/2 is √6/4.

1/2 times √2/2 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 105° = (√6 + √2)/4.

sin 105° = √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 sin 105° on the calculator is about 0.9659.

The same!

So it is right.

sin 105° ≈ 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