Step by step
The question
Question: find the exact value of sin 75°.
Exact means with roots and fractions, not a decimal.
1 Split the angle
The angle is already split: 45° + 30°.
Both are angles from the unit circle.
So sin 75° is sin of 45° plus 30°.
2 Write the rule
The sum rule for sine: sin(A + B) = sin A cos B + cos A sin B.
Careful!
We cannot split sin into two sins.
sin 45° + sin 30° is about 1.2071: more than 1.
A sine is never more than 1.
We use the rule.
A is 45° and B is 30°.
We put them in the rule.
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.
We put the values in.
4 Multiply
Tops times tops, bottoms times bottoms.
A root times a root: √2 × √3 = √6.
√2/2 times √3/2 is √6/4.
√2/2 times 1/2 is √2/4.
5 Make one fraction
Both have the same bottom, 4.
So we put the tops together over 4: (√6 + √2)/4.
So sin 75° = (√6 + √2)/4.
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.
And sin 75° on the calculator is about 0.9659.
The same!
So it is right.
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.