Step by step
The question
Question: find angle A of the triangle with sides b = 5, c = 6 and a = 4.
1 Write the law of cosines
We know all three sides.
We want angle A, across from side a.
The law of cosines: a² = b² + c² − 2bc cos A.
Side a is across from angle A.
It is Pythagoras with a correction.
When A is 90°, cos 90° is 0, and it is just Pythagoras.
2 Put the numbers in
Now put the numbers in: a = 4, b = 5 and c = 6.
Squares first: 16, 25 and 36.
And 2 × 5 × 6 is 60.
Careful!
60 cos A is one piece: 60 times cos A.
We cannot take 60 from 61 first.
25 + 36 is 61.
We swap the two sides, so cos A is on the left.
We want cos A by itself.
Two things are stuck to cos A: times −60 and + 61.
The + 61 is on the outside, so it comes off first.
Then the times −60.
3 Remove the 61
To remove 61, we take away 61.
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.
61 and − 61 make zero.
Cross them out.
16 − 61 = −45.
Now we have −60 cos A = −45.
4 Remove the −60
−60 cos A means −60 times cos A.
We want just cos A.
To remove times −60, we divide by −60.
We do it on both sides, so they stay equal.
−60 divided by −60 is 1.
So just cos A is left.
We can make −45 over −60 simpler.
It is the same as 3/4.
So cos A = 3/4!
As a decimal: 0.7500.
5 Turn it into the angle
The cos⁻¹ button turns the ratio back into the angle.
The calculator gives about 41.41.
To the nearest tenth, A ≈ 41.4°.
6 Check
Let's check: put A back.
61 take away 60 times cos 41.4° is about 16.0.
And a squared is 16.
The same!
So it is right.
Let's look back.
First: write the law of cosines.
Then: put the numbers in.
Then: remove the 61.
Then: remove the −60.
Then: turn it into the angle.
Last: check.