Step by step
The question
Question: find angle A of the triangle with sides b = 7, c = 6 and a = 10.
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 = 10, b = 7 and c = 6.
Squares first: 100, 49 and 36.
And 2 × 7 × 6 is 84.
Careful!
84 cos A is one piece: 84 times cos A.
We cannot take 84 from 85 first.
49 + 36 is 85.
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 −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.
100 − 85 = 15.
Now we have −84 cos A = 15.
4 Remove the −84
−84 cos A means −84 times cos A.
We want just cos A.
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 A is left.
We can make 15 over −84 simpler.
It is the same as −5/28.
So cos A = −5/28!
As a decimal, about −0.1786.
5 Turn it into the angle
The cos⁻¹ button turns the ratio back into the angle.
The calculator gives about 100.287.
To the nearest tenth, A ≈ 100.3°.
6 Check
Let's check: put A back.
85 take away 84 times cos 100.3° is about 100.0.
And a squared is 100.
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.