Step by step
The question
Question: find side c of the triangle with A = 60°, C = 35° and b = 20.
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 Find the third angle
We need angle B.
The angles of a triangle add to 180°.
So B is 180° take away the other two.
So B = 85°.
3 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.
4 Put the numbers in
Now put the numbers in: C = 35°, b = 20 and B = 85°.
5 Find c
c is on top, over sin 35°.
To remove the sin 35° at the bottom, we multiply by sin 35°.
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 35° over sin 35° is 1.
So just c is left.
So c = 20 × sin 35° over sin 85°.
The calculator gives sin 35°: about 0.5736.
And sin 85°: about 0.9962.
Put the numbers in.
That is about 11.515.
To the nearest tenth, c ≈ 11.5.
6 Check
Let's check: both pairs must give the same number.
c over sin 35° is about 20.05.
b over sin 85° is about 20.08.
Almost the same: the gap is only from rounding.
And the bigger angle has the bigger side: 35° is less than 85°, and 11.5 is less than 20.
So it is right!
Let's look back.
First: match sides and angles.
Then: find the third angle.
Then: write the law of sines.
Then: put the numbers in.
Then: find c.
Last: check.