Step by step
The question
Question: find side b of the triangle with a = 12, c = 10 and B = 35°.
1 Write the law of cosines
We know two sides, a and c, and the angle B between them.
The law of cosines: b² = a² + c² − 2ac cos B.
Side b is across from angle B.
It is Pythagoras with a correction.
When B is 90°, cos 90° is 0, and it is just Pythagoras.
2 Put the numbers in
Now put the numbers in: a = 12, c = 10 and B = 35°.
3 Work it out
Squares first: 12² is 144, 10² is 100.
And 2 × 12 × 10 is 240.
Careful!
Times comes before take away.
Work out 240 × cos 35° first, then take it away.
The calculator gives cos 35°: about 0.8192.
240 times that is about 196.61.
144 + 100 is 244.
So b² ≈ 47.39.
4 Take the square root
b squared is 47.39.
So b is the square root of 47.39.
A side is never negative, so only the plus root.
The calculator gives about 6.884.
To the nearest tenth, b ≈ 6.9.
5 Check
Let's check: the law turned around gives cos B back.
244 take away 47.39, over 240: 0.8192.
And cos 35° is about 0.8192.
The same!
So b 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.