Step by step
The question
Question: find angle C of the triangle with sides a = 3, b = 5 and c = 7.
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 = 7, a = 3 and b = 5.
Squares first: 49, 9 and 25.
And 2 × 3 × 5 is 30.
Careful!
30 cos C is one piece: 30 times cos C.
We cannot take 30 from 34 first.
9 + 25 is 34.
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 −30 and + 34.
The + 34 is on the outside, so it comes off first.
Then the times −30.
3 Remove the 34
To remove 34, we take away 34.
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.
34 and − 34 make zero.
Cross them out.
49 − 34 = 15.
Now we have −30 cos C = 15.
4 Remove the −30
−30 cos C means −30 times cos C.
We want just cos C.
To remove times −30, we divide by −30.
We do it on both sides, so they stay equal.
−30 divided by −30 is 1.
So just cos C is left.
We can make 15 over −30 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.
34 take away 30 times cos 120° is about 49.0.
And c squared is 49.
The same!
So it is right.
Let's look back.
First: write the law of cosines.
Then: put the numbers in.
Then: remove the 34.
Then: remove the −30.
Then: turn it into the angle.
Last: check.