Step by step
The question
Question: find side c of the triangle with a = 8, b = 11 and C = 40°.
1 Write the law of cosines
We know two sides, a and b, and the angle C between them.
The law of cosines: c² = a² + b² − 2ab cos C.
Side c is across from angle C.
It is Pythagoras with a correction.
When C is 90°, cos 90° is 0, and it is just Pythagoras.
2 Put the numbers in
Now put the numbers in: a = 8, b = 11 and C = 40°.
3 Work it out
Squares first: 8² is 64, 11² is 121.
And 2 × 8 × 11 is 176.
Careful!
Times comes before take away.
Work out 176 × cos 40° first, then take it away.
The calculator gives cos 40°: about 0.7660.
176 times that is about 134.82.
64 + 121 is 185.
So c² ≈ 50.18.
4 Take the square root
c squared is 50.18.
So c is the square root of 50.18.
A side is never negative, so only the plus root.
The calculator gives about 7.084.
To the nearest tenth, c ≈ 7.1.
5 Check
Let's check: the law turned around gives cos C back.
185 take away 50.18, over 176: 0.766.
And cos 40° is about 0.7660.
The same!
So c is right.
Let's look back.
First: write the law of cosines.
Then: put the numbers in.
Then: work it out.
Then: take the square root.
Last: check.