Step by step
The question
Question: find angle C of the triangle with sides a = 7, b = 8 and c = 13.
1 Write the law of cosines
We know all three sides.
We want angle C, across from side c.
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: c = 13, a = 7 and b = 8.
Squares first: 169, 49 and 64.
And 2 × 7 × 8 is 112.
Careful!
112 cos C is one piece: 112 times cos C.
We cannot take 112 from 113 first.
49 + 64 is 113.
We swap the two sides, so cos C is on the left.
We want cos C by itself.
Two things are stuck to cos C: times −112 and + 113.
The + 113 is on the outside, so it comes off first.
Then the times −112.
3 Remove the 113
To remove 113, we take away 113.
We do it on both sides.
Why both sides?
The = sign means both sides are the same amount.
If we change only one side, they are not the same any more.
113 and − 113 make zero.
Cross them out.
169 − 113 = 56.
Now we have −112 cos C = 56.
4 Remove the −112
−112 cos C means −112 times cos C.
We want just cos C.
To remove times −112, we divide by −112.
We do it on both sides, so they stay equal.
−112 divided by −112 is 1.
So just cos C is left.
We can make 56 over −112 simpler.
It is the same as −1/2.
So cos C = −1/2!
As a decimal: −0.5000.
5 Turn it into the angle
The cos⁻¹ button turns the ratio back into the angle.
The calculator gives exactly 120°.
So C = 120°.
6 Check
Let's check: put C back.
113 take away 112 times cos 120° is about 169.0.
And c squared is 169.
The same!
So it is right.
Let's look back.
First: write the law of cosines.
Then: put the numbers in.
Then: remove the 113.
Then: remove the −112.
Then: turn it into the angle.
Last: check.