Step by step
The question
Question: find angle C of the triangle with sides a = 5, b = 7 and c = 8.
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 = 8, a = 5 and b = 7.
Squares first: 64, 25 and 49.
And 2 × 5 × 7 is 70.
Careful!
70 cos C is one piece: 70 times cos C.
We cannot take 70 from 74 first.
25 + 49 is 74.
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 −70 and + 74.
The + 74 is on the outside, so it comes off first.
Then the times −70.
3 Remove the 74
To remove 74, we take away 74.
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.
74 and − 74 make zero.
Cross them out.
64 − 74 = −10.
Now we have −70 cos C = −10.
4 Remove the −70
−70 cos C means −70 times cos C.
We want just cos C.
To remove times −70, we divide by −70.
We do it on both sides, so they stay equal.
−70 divided by −70 is 1.
So just cos C is left.
We can make −10 over −70 simpler.
It is the same as 1/7.
So cos C = 1/7!
As a decimal, about 0.1429.
5 Turn it into the angle
The cos⁻¹ button turns the ratio back into the angle.
The calculator gives about 81.787.
To the nearest tenth, C ≈ 81.8°.
6 Check
Let's check: put C back.
74 take away 70 times cos 81.8° is about 64.0.
And c squared is 64.
The same!
So it is right.
Let's look back.
First: write the law of cosines.
Then: put the numbers in.
Then: remove the 74.
Then: remove the −70.
Then: turn it into the angle.
Last: check.