This article originally appeared in Dev.Mag Issue 14, released in June 2007

Trigonometry. The mere mention of it strikes terror into the minds of bewildered high school students. Confusing ratios, bizarre diagrams, trying to find the sine of the cosine of the tangent of theta - many people struggle to wrap their minds around it and eventually give up in desperation to do something easier, such as ending war or achieving cold fusion.

Well, despite what your high school teachers may have said, trigonometry can actually be useful! Yes, in the vast sea of unintelligible terms is a valuable and powerful mathematical tool that can aid you immensely in the humble task of game design. The aim of this series is to explain to you just how simple trigonometry really is, and how you can use it to solve some very common programming problems. Hopefully it will make the job of coding your next blockbuster game just that little bit easier.

First Principles -The Triangle:

Before we start writing any trig in our code, it’s necessary to understand all the nasty mathematical stuff behind it (just joking, it’s not that bad!). First, we should take a look at how trigonometry works at the most basic level. To do that, we’ll need a triangle. More specifically, we’ll need a right-angled triangle (a triangle that contains a single 90 degree corner). Here’s one I found lying around:

A boring right-angled triangle

Firstly, it’s important that we name the sides of the triangle so that we can identify them when we build our equations later. In trig, we name sides based on their relationships to the corners of the triangle. Take this triangle as an example:

Boring right-angled triangle with angle theta

In this triangle, we’ve defined one of the non-right-angles with the variable theta (θ). For now, this is simply a variable to identify the corner - don’t worry about about how the numbers fit in just yet!

Now that we’ve defined θ, we can use it to identify our sides.

Right-angled triangle with angles and sides defined

Now we’ve designated our sides. In trig, sides are always designated relative to the angle that we’re working with, θ in this case. o is the side opposite to the specified angle. a is the angle adjacent (next to) the specified angle. h is the hypotenuse, which is always the side opposite to the 90 degree angle. Clear? Well, if not, let’s go through the same exercise with the other angle in the triangle. Let’s designate that one Alpha (α), and take a look at how this affects our sides.

Right-angled triangle with angle alpha, and sides

Note that this triangle is still the same triangle. We’ve just renamed the sides according to which corner we’re viewing them relative to, in exactly the same way that we did with angle θ. You’ll note that h is still assigned to the same side. As I mentioned, this is because the hypotenuse is always the side opposite to the 90 degree angle, and so won’t change if we switch corners. If this still seems a little confusing, don’t worry - next we’ll be seeing how all of this ties together!