Step by step
The question
Question: find side c of the triangle with a = 5, b = 7 and C = 60°.
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 = 7 and C = 60°.
3 Work it out
Squares first: 5² is 25, 7² is 49.
And 2 × 5 × 7 is 70.
Careful!
Times comes before take away.
Work out 70 × cos 60° first, then take it away.
The calculator gives cos 60°: exactly 0.5.
70 times that is 35.
25 + 49 is 74.
So c² = 39.
4 Take the square root
c squared is 39.
So c is the square root of 39.
A side is never negative, so only the plus root.
The calculator gives about 6.245.
To the nearest tenth, c ≈ 6.2.
5 Check
Let's check: the law turned around gives cos C back.
74 take away 39, over 70: 0.5.
And cos 60° is 0.5.
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.