Step by step
The question
Question: find the exact value of sin 165°.
Exact means with roots and fractions, not a decimal.
1 Split the angle
165° is not on the unit circle.
But 120° and 45° are, and 120° plus 45° is 165°.
So sin 165° is sin of 120° plus 45°.
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 120° + sin 45° is about 1.5731: more than 1.
A sine is never more than 1.
We use the rule.
A is 120° and B is 45°.
We put them in the rule.
3 Put in the values
From the unit circle: sin 120° = √3/2.
cos 45° = √2/2.
cos 120° = −1/2.
sin 45° = √2/2.
120° is in quadrant II: the signs come from there.
We put the values in.
4 Multiply
Tops times tops, bottoms times bottoms.
A root times a root: √3 × √2 = √6.
√3/2 times √2/2 is √6/4.
−1/2 times √2/2 is −√2/4.
Adding a negative is taking away.
5 Make one fraction
Both have the same bottom, 4.
So we put the tops together over 4: (√6 − √2)/4.
So sin 165° = (√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 0.2588.
And sin 165° on the calculator is about 0.2588.
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.