Step by step
The question
Question: find the exact value of cos 105°.
Exact means with roots and fractions, not a decimal.
1 Split the angle
105° is not on the unit circle.
But 60° and 45° are, and 60° plus 45° is 105°.
So cos 105° is cos of 60° plus 45°.
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 60° and B is 45°.
We put them in the rule.
3 Put in the values
From the unit circle: cos 60° = 1/2.
cos 45° = √2/2.
sin 60° = √3/2.
sin 45° = √2/2.
We put the values in.
4 Multiply
Tops times tops, bottoms times bottoms.
A root times a root: √3 × √2 = √6.
1/2 times √2/2 is √2/4.
√3/2 times √2/2 is √6/4.
5 Make one fraction
Both have the same bottom, 4.
So we put the tops together over 4: (√2 − √6)/4.
So cos 105° = (√2 − √6)/4.
6 Check
Let's check with the calculator.
Our answer: √2 is about 1.4142, √6 is about 2.4495.
Over 4: about −0.2588.
And cos 105° 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.