Step by step
The question
Question: find side c of the triangle with A = 55°, C = 65° and a = 7 cm.
1 Match sides and angles
Each side has a partner: the angle across from it.
Side a is across from angle A.
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 = a over sin A.
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 = 65°, a = 7 and A = 55°.
4 Find c
c is on top, over sin 65°.
To remove the sin 65° at the bottom, we multiply by sin 65°.
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 65° over sin 65° is 1.
So just c is left.
So c = 7 × sin 65° over sin 55°.
The calculator gives sin 65°: about 0.9063.
And sin 55°: about 0.8192.
Put the numbers in.
That is about 7.745.
To the nearest tenth, c ≈ 7.7 cm.
5 Check
Let's check: both pairs must give the same number.
c over sin 65° is about 8.50.
a over sin 55° is about 8.55.
Almost the same: the gap is only from rounding.
And the bigger angle has the bigger side: 65° is more than 55°, and 7.7 is more than 7.
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.