Step by step
The question
Question: find side c of the triangle with B = 35°, C = 80° and b = 9.
1 Match sides and angles
Each side has a partner: the angle across from it.
Side b is across from angle B.
Side c, the one we want, is across from angle C.
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: c over sin C = 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: C = 80°, b = 9 and B = 35°.
4 Find c
c is on top, over sin 80°.
To remove the sin 80° at the bottom, we multiply by sin 80°.
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 80° over sin 80° is 1.
So just c is left.
So c = 9 × sin 80° over sin 35°.
The calculator gives sin 80°: about 0.9848.
And sin 35°: about 0.5736.
Put the numbers in.
That is about 15.453.
To the nearest tenth, c ≈ 15.5.
5 Check
Let's check: both pairs must give the same number.
c over sin 80° is about 15.74.
b over sin 35° is about 15.69.
Almost the same: the gap is only from rounding.
And the bigger angle has the bigger side: 80° is more than 35°, and 15.5 is more than 9.
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 c.
Last: check.