Step by step
The question
Question: find angle A of the triangle with sides b = 7, c = 5 and a = 9.
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 = 9, b = 7 and c = 5.
Squares first: 81, 49 and 25.
And 2 × 7 × 5 is 70.
Careful!
70 cos A is one piece: 70 times cos A.
We cannot take 70 from 74 first.
49 + 25 is 74.
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 −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.
81 − 74 = 7.
Now we have −70 cos A = 7.
4 Remove the −70
−70 cos A means −70 times cos A.
We want just cos A.
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 A is left.
We can make 7 over −70 simpler.
It is the same as −1/10.
So cos A = −1/10!
As a decimal: −0.1000.
5 Turn it into the angle
The cos⁻¹ button turns the ratio back into the angle.
The calculator gives about 95.739.
To the nearest tenth, A ≈ 95.7°.
6 Check
Let's check: put A back.
74 take away 70 times cos 95.7° is about 81.0.
And a squared is 81.
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.