In case you’re not familiar with what this means, a variable is an unknown quantity that we might be able to find out more about using its relationships to other variables and constants. For example, 5 - x = 3has the solution x = 2, but other variable equations might have 0 or many solutions, such as x = (5 - 4) x, which has all numbers as a solution for x. But often times in trigonometry, we might not even care about finding a solution in terms of a number like x = 2, but rather, some solution in terms of more variables, like x = y. Let’s learn the ball game for trigonometry.  

Angle and Side Notation

 Let’s first discuss some terminology that applies to all of geometry and beyond: edges are the sides of a triangle (just a more fancy term because mathematicians like to feel fancy-schmancy), and vertices (singular: vertex) are the points where two edges meet.

Pop Quiz! How many edges does a triangle have? How many vertices does it have?

(Answer: 3 and 3, by counting)

Variable Naming

Now, let’s take a look at how variable names factor into all of this.
Most commonly, mathematicians will give the vertices names, and leave it at that, like so:

Now, you might say, “Wow, how lazy are these mathematicians? Not even giving the triangle, its edges, or even its angles a name!”

And yes, I agree with you: mathematicians are generally pretty lazy, but I swear there’s a good reason for these things! Generally, we just make do with what we have. For example, each edge is in between two vertices, right? So why not call the edge in between A and B AB(order doesn’t really matter here), such that we get this: 

So, essentially, we give the edges the most obvious name possible. 

How about the angles? Well, here, there are two conventions: 1. use three vertices to name the angle, or 2. (my favorite) use one vertex.
To explain the first convention, we have to see that each angle is directly sandwiched between two vertices:

See that angle at vertex A? We can follow the vertices we used to make this angle (B, then A, then C), and call the angle∠BAC. (Exercise: use this convention to name all angles)
Or, you could simply call the angle by the vertex that it is on (∠A for the angle above) and then call it a day.

 Finally, we call the triangle ΔABC, which I think is the simplest of all.

Homework questions:

How do we quickly convert between the triple vertex angle convention and single vertex angle convention?

(From the triple vertex one (e.g., BAC), and call the angle by the middle letter (A); from the single vertex one (A), look at the other two vertex names and sandwich the single vertex (BAC))

Use the first angle convention to name all the angles in ABCabove.