Step by step
The question
Question: find the exact value of cos 5π/12.
Exact means with roots and fractions, not a decimal.
1 Split the angle
5π/12 is not on the unit circle.
But π/4 and π/6 are, and π/4 plus π/6 is 5π/12.
So cos 5π/12 is cos of π/4 plus π/6.
2 Write the rule
The sum rule for cosine: cos(A + B) = cos A cos B − sin A sin B.
Look at the sign in the middle: it is the other one, minus.
Careful!
For cosine the sign in the middle flips.
A plus inside gives minus in the middle.
A is π/4 and B is π/6.
We put them in the rule.
3 Put in the values
From the unit circle: cos π/4 = √2/2.
cos π/6 = √3/2.
sin π/4 = √2/2.
sin π/6 = 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 cos 5π/12 = (√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 cos 5π/12 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.