Step by step
The question
Question: find side b of the triangle with A = 40°, B = 60° and a = 8.
1 Match sides and angles
Each side has a partner: the angle across from it.
Side a is across from angle A.
Side b, the one we want, is across from angle B.
2 Write the law of sines
In every triangle, a side over the sine of its partner angle gives the same number, for all three pairs.
We use the pair we want and the pair we know: b over sin B = a over sin A.
Careful!
Each side goes with the angle across from it, never with an angle next to it.
3 Put the numbers in
Now put the numbers in: B = 60°, a = 8 and A = 40°.
4 Find b
b is on top, over sin 60°.
To remove the sin 60° at the bottom, we multiply by sin 60°.
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.
sin 60° over sin 60° is 1.
So just b is left.
So b = 8 × sin 60° over sin 40°.
The calculator gives sin 60°: about 0.8660.
And sin 40°: about 0.6428.
Put the numbers in.
That is about 10.778.
To the nearest tenth, b ≈ 10.8.
5 Check
Let's check: both pairs must give the same number.
b over sin 60° is about 12.47.
a over sin 40° is about 12.45.
Almost the same: the gap is only from rounding.
And the bigger angle has the bigger side: 60° is more than 40°, and 10.8 is more than 8.
So it is right!
Let's look back.
First: match sides and angles.
Then: write the law of sines.
Then: put the numbers in.
Then: find b.
Last: check.