Step by step
The question
Question: find side a of the triangle with A = 30°, B = 90° and b = 14.
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 = 30°, b = 14 and B = 90°.
4 Find a
a is on top, over sin 30°.
To remove the sin 30° at the bottom, we multiply by sin 30°.
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 30° over sin 30° is 1.
So just a is left.
So a = 14 × sin 30° over sin 90°.
The calculator gives sin 30°: exactly 0.5.
And sin 90°: exactly 1.
Put the numbers in.
That is 7.
So a = 7.
5 Check
Let's check: both pairs must give the same number.
a over sin 30° is about 14.00.
b over sin 90° is 14.00 too.
The same!
And the bigger angle has the bigger side: 30° is less than 90°, and 7 is less than 14.
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.