Step by step
The question
Question: find angle C of the triangle with sides a = 6, b = 8 and c = 10.
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 = 10, a = 6 and b = 8.
Squares first: 100, 36 and 64.
And 2 × 6 × 8 is 96.
Careful!
96 cos C is one piece: 96 times cos C.
We cannot take 96 from 100 first.
36 + 64 is 100.
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 −96 and + 100.
The + 100 is on the outside, so it comes off first.
Then the times −96.
3 Remove the 100
To remove 100, we take away 100.
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.
100 and − 100 make zero.
Cross them out.
100 − 100 = 0.
Now we have −96 cos C = 0.
4 Remove the −96
−96 cos C means −96 times cos C.
We want just cos C.
To remove times −96, we divide by −96.
We do it on both sides, so they stay equal.
−96 divided by −96 is 1.
So just cos C is left.
0 divided by −96 is 0.
So cos C = 0!
As a decimal: 0.0000.
5 Turn it into the angle
The cos⁻¹ button turns the ratio back into the angle.
The calculator gives exactly 90°.
So C = 90°.
6 Check
Let's check: put C back.
100 take away 96 times cos 90° is about 100.0.
And c squared is 100.
The same!
So it is right.
Let's look back.
First: write the law of cosines.
Then: put the numbers in.
Then: remove the 100.
Then: remove the −96.
Then: turn it into the angle.
Last: check.