Step by step
The question
Question: find side b of the triangle with A = 52°, B = 71° and a = 9.5.
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.
Why is it true?
Draw the height h from corner C, straight down to the side across from it.
It makes two right triangles.
In one, h is b times sin A.
In the other, h is a times sin B.
Both are h, so they are equal.
Divide both sides by sin B times sin A.
Each side is left over the sine of its partner angle.
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 = 71°, a = 9.5 and A = 52°.
4 Find b
b is on top, over sin 71°.
To remove the sin 71° at the bottom, we multiply by sin 71°.
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 71° over sin 71° is 1.
So just b is left.
So b = 9.5 × sin 71° over sin 52°.
The calculator gives sin 71°: about 0.9455.
And sin 52°: about 0.7880.
Put the numbers in.
That is about 11.399.
To the nearest tenth, b ≈ 11.4.
5 Check
Let's check: both pairs must give the same number.
b over sin 71° is about 12.06.
a over sin 52° is 12.06 too.
The same!
And the bigger angle has the bigger side: in every triangle the biggest angle faces the longest side.
71° is more than 52°, and 11.4 is more than 9.5.
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.