Step by step
The question
Question: find the exact value of cos 75°.
Exact means with roots and fractions, not a decimal.
1 Split the angle
75° is not on the unit circle.
But 45° and 30° are, and 45° plus 30° is 75°.
So cos 75° is cos of 45° plus 30°.
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 45° and B is 30°.
We put them in the rule.
3 Put in the values
From the unit circle: cos 45° = √2/2.
cos 30° = √3/2.
sin 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 cos 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 0.2588.
And cos 75° 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.