Step by step
The question
Question: find angle C of the triangle with sides a = 7, b = 8 and c = 9.
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.
Where does it come from?
Put corner C at 0, with side a along the x-axis.
The last corner is b times the unit-circle point: b cos C across, b sin C up.
Side c joins two corners.
Pythagoras on the across and up gaps gives c squared.
Open the brackets.
Take b² out of the two square parts.
cos squared plus sin squared is 1.
That leaves the law.
When C is 90°, cos 90° is 0, and it is just Pythagoras.
2 Put the numbers in
Now put the numbers in: c = 9, a = 7 and b = 8.
Squares first: 81, 49 and 64.
And 2 × 7 × 8 is 112.
Careful!
112 cos C is one piece: 112 times cos C.
We cannot take 112 from 113 first.
49 + 64 is 113.
We swap the two sides, so cos C is on the left.
We want cos C by itself.
First we remove the + 113, so −112 cos C is alone.
Then we remove the times −112.
3 Remove the 113
To remove 113, we take away 113.
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.
113 and − 113 cancel each other.
They make zero.
81 − 113 = −32.
Now we have −112 cos C = −32.
So after step 3: we took away 113, and now we have −112 cos C = −32.
4 Remove the −112
−112 cos C means −112 times cos C.
To remove times −112, we divide by −112.
We do it on both sides, so they stay equal.
−112 divided by −112 is 1.
They cancel each other.
So just cos C is left.
We can make −32 over −112 simpler.
It is the same as 2/7.
So cos C = 2/7!
As a decimal, about 0.2857.
So after step 4: we divided by 112, and now we have cos C ≈ 0.2857.
5 Turn it into the angle
The cos⁻¹ button turns the ratio back into the angle.
The calculator gives about 73.398.
To the nearest tenth, C ≈ 73.4°.
6 Check
Let's check: put C back.
113 take away 112 times cos 73.4° is about 81.0.
And c 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 113.
Then: remove the −112.
Then: turn it into the angle.
Last: check.