Step by step
The question
Question: find side b of the triangle with A = 30°, B = 45° and a = 10.
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 = 45°, a = 10 and A = 30°.
4 Find b
b is on top, over sin 45°.
To remove the sin 45° at the bottom, we multiply by sin 45°.
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 45° over sin 45° is 1.
So just b is left.
So b = 10 × sin 45° over sin 30°.
The calculator gives sin 45°: about 0.7071.
And sin 30°: exactly 0.5.
Put the numbers in.
That is about 14.142.
To the nearest tenth, b ≈ 14.1.
5 Check
Let's check: both pairs must give the same number.
b over sin 45° is about 19.94.
a over sin 30° is about 20.00.
Almost the same: the gap is only from rounding.
And the bigger angle has the bigger side: 45° is more than 30°, and 14.1 is more than 10.
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.