Triangles teach trig.
Right angles always around.
Old bonds break away.
Yep, we've starting things off with a haiku, and there's nothing you can do to stop us. Now we're going to use trig on non-right triangles, and finally lift the right-angle restriction we've been operating under since day 1.
So, let's look at a non-right triangle, shall we?
What a-cute triangle. To understand non-right triangles and how we can use trig with them, we need to create a few right triangles within this guy. We'll be done comparing them to right triangles soon, we promise.
Let's start investigating the height, h, using these right triangles. We did this a bit last time, when we found the new area formula for triangles.
h = c sin A
We're not stopping here, though. We can find h another way, using C.
h = a sin C
Two we set these two equal, because we can totally do that.
c sin A = a sin C
We now have a trig equation relating sides and angles of a non-right triangle. But B is feeling left out. We can fix that, though, by using the height g just like we did h.
Which, with some rearranging gives us:
See how our assistant, 
, has shown up a second time? With its help, we're ready for the grand reveal, the Law of Sines:
Ta-da. This says that the proportions of each angle and its opposite side are equal throughout a triangle. We can now use trig to find missing sides or angles from all kinds of triangles, not just right triangles. Think of it as our eye in the sky searchlight when hunting for a missing piece.
Sample Problem
A triangle has a 65° angle with the side opposite that angle measuring 18 units. What is the measurement of the side opposite a 32° angle in the same triangle?
The Law of Sines shows us the way. We only need two of the equalities, not all three.
We keep each side and angle paired together: a with A, and b with B. We can then rearrange things to solve for b:
But, uh, how are we going to deal with those angles? Double angles, half angles, none of them look any good. Well, we're breaking our unspoken rule and giving out some non-standard angles now that we're in the last section. Yep, dust off those calculators, make sure they're set for "Degree" mode, and get ready to calculate.
= 10.52 units
Using the calculator again felt good. We missed you buddy.
Sometimes we'll be looking for an unknown angle instead of a side. We're lazy, and so to avoid an extra step, we can rewrite the Law of Sines by dividing the whole thing by 1.
This is the same formula, just flippy-flopped upside down. Just don't drop it while spinning it around.