Step by step
The question
Question: find angle B of the triangle with sides a = 8 cm, c = 5 cm and b = 6 cm.
1 Write the law of cosines
We know all three sides.
We want angle B, across from side b.
The law of cosines: b² = a² + c² − 2ac cos B.
Side b is across from angle B.
It is Pythagoras with a correction.
When B is 90°, cos 90° is 0, and it is just Pythagoras.
2 Put the numbers in
Now put the numbers in: b = 6, a = 8 and c = 5.
Squares first: 36, 64 and 25.
And 2 × 8 × 5 is 80.
Careful!
80 cos B is one piece: 80 times cos B.
We cannot take 80 from 89 first.
64 + 25 is 89.
We swap the two sides, so cos B is on the left.
We want cos B by itself.
Two things are stuck to cos B: times −80 and + 89.
The + 89 is on the outside, so it comes off first.
Then the times −80.
3 Remove the 89
To remove 89, we take away 89.
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.
89 and − 89 make zero.
Cross them out.
36 − 89 = −53.
Now we have −80 cos B = −53.
4 Remove the −80
−80 cos B means −80 times cos B.
We want just cos B.
To remove times −80, we divide by −80.
We do it on both sides, so they stay equal.
−80 divided by −80 is 1.
So just cos B is left.
We can make −53 over −80 simpler.
It is the same as 53/80.
So cos B = 53/80!
As a decimal: 0.6625.
5 Turn it into the angle
The cos⁻¹ button turns the ratio back into the angle.
The calculator gives about 48.509.
To the nearest tenth, B ≈ 48.5°.
6 Check
Let's check: put B back.
89 take away 80 times cos 48.5° is about 36.0.
And b squared is 36.
The same!
So it is right.
Let's look back.
First: write the law of cosines.
Then: put the numbers in.
Then: remove the 89.
Then: remove the −80.
Then: turn it into the angle.
Last: check.