Step by step
The question
Question: find side c of the triangle with a = 5, b = 8 and C = 25°.
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 = 5, b = 8 and C = 25°.
3 Work it out
Squares first: 5² is 25, 8² is 64.
And 2 × 5 × 8 is 80.
Careful!
Times comes before take away.
Work out 80 × cos 25° first, then take it away.
The calculator gives cos 25°: about 0.9063.
80 times that is about 72.5.
25 + 64 is 89.
So c² ≈ 16.5.
4 Take the square root
c squared is 16.5.
So c is the square root of 16.5.
A side is never negative, so only the plus root.
The calculator gives about 4.062.
To the nearest tenth, c ≈ 4.1.
5 Check
Let's check: the law turned around gives cos C back.
89 take away 16.5, over 80: 0.9063.
And cos 25° is about 0.9063.
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.