Over in the Mental Gym, there's a tutorial all about the unit circle, which I published back in February. It focuses mainly on radians, Pi, sine and cosine.

I've had a few e-mails asking how to understand the other trigonometric functions: tangent, cotangent, secant, and cosecant. As of today, that new tutorial is up - Unit Circle 2: Trig Functions.

As suggested on the first page of the new tutorial, you should read and understand the original unit circle tutorial first.

In high school trigonometry, most people are told they'll really only need to know about sine and cosine, and a little bit about tangent. The formulas are explained, but never really their nature. It's one thing to be able to get the answer, and quite another to understand what the answer means.

My goal in the new tutorial was to be clear about the whats and whys of the other trig functions. You may never use them, but if you understand them, you'll quickly recognize when you can potentially use them.

Once you understand what each function covers, I lock in the important points of each function with mnemonics. Sure, you've probably run across SOHCAHTOA, but I've got a less familiar mnemonic covering all 6 functions that will have you saying “OOH! AAH!”.

Thanks to my recent experiences with Wolfram|Alpha, links to their knowledge engine were used in several examples. I only wish I'd known about Wolfram|Alpha sooner!

Considering the number of books and articles through which I had to look to develop these descriptions, I do believe that this new tutorial presents tangents, cotangents, secants, and cosecants in a way that's among the clearest out there.

If you have any further suggestions about the clarity of this tutorial (or anything else on Grey Matters), or have any questions about something that could be presented more clearly, let me know in the comments!