Step by step
The question
Question: find side a of the triangle with A = 50°, C = 70° and c = 12 cm.
1 Match sides and angles
Each side has a partner: the angle across from it.
Side c is across from angle C.
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 = c over sin C.
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 = 50°, c = 12 and C = 70°.
4 Find a
a is on top, over sin 50°.
To remove the sin 50° at the bottom, we multiply by sin 50°.
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 50° over sin 50° is 1.
So just a is left.
So a = 12 × sin 50° over sin 70°.
The calculator gives sin 50°: about 0.7660.
And sin 70°: about 0.9397.
Put the numbers in.
That is about 9.782.
To the nearest tenth, a ≈ 9.8 cm.
5 Check
Let's check: both pairs must give the same number.
a over sin 50° is about 12.79.
c over sin 70° is about 12.77.
Almost the same: the gap is only from rounding.
And the bigger angle has the bigger side: 50° is less than 70°, and 9.8 is less 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.