Step by step
The question
Question: find side a of the triangle with A = 70°, B = 50° and b = 12.
1 Match sides and angles
Each side has a partner: the angle across from it.
Side b is across from angle B.
Side a, the one we want, is across from angle A.
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: a over sin A = b over sin B.
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: A = 70°, b = 12 and B = 50°.
4 Find a
a is on top, over sin 70°.
To remove the sin 70° at the bottom, we multiply by sin 70°.
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 70° over sin 70° is 1.
So just a is left.
So a = 12 × sin 70° over sin 50°.
The calculator gives sin 70°: about 0.9397.
And sin 50°: about 0.7660.
Put the numbers in.
That is about 14.72.
To the nearest tenth, a ≈ 14.7.
5 Check
Let's check: both pairs must give the same number.
a over sin 70° is about 15.64.
b over sin 50° is about 15.66.
Almost the same: the gap is only from rounding.
And the bigger angle has the bigger side: 70° is more than 50°, and 14.7 is more than 12.
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 a.
Last: check.