Step by step
The question
Question: find angle C of the triangle with sides a = 6, b = 7 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.
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 = 9, a = 6 and b = 7.
Squares first: 81, 36 and 49.
And 2 × 6 × 7 is 84.
Careful!
84 cos C is one piece: 84 times cos C.
We cannot take 84 from 85 first.
36 + 49 is 85.
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 −84 and + 85.
The + 85 is on the outside, so it comes off first.
Then the times −84.
3 Remove the 85
To remove 85, we take away 85.
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.
85 and − 85 make zero.
Cross them out.
81 − 85 = −4.
Now we have −84 cos C = −4.
4 Remove the −84
−84 cos C means −84 times cos C.
We want just cos C.
To remove times −84, we divide by −84.
We do it on both sides, so they stay equal.
−84 divided by −84 is 1.
So just cos C is left.
We can make −4 over −84 simpler.
It is the same as 1/21.
So cos C = 1/21!
As a decimal, about 0.0476.
5 Turn it into the angle
The cos⁻¹ button turns the ratio back into the angle.
The calculator gives about 87.271.
To the nearest tenth, C ≈ 87.3°.
6 Check
Let's check: put C back.
85 take away 84 times cos 87.3° 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 85.
Then: remove the −84.
Then: turn it into the angle.
Last: check.