The “Complete Triangle” figure —which represents the six trig functions as lengths of segments— is something of an obsession of mine, and has been for some time.
It’s the basis of my triple-ricochet company logo and name of my iOS-specific brand, tricochet, and it’s the subject of my very first iOS app (one of the very first in the App Store when it opened), Trigger … which is actually the latest evolution of what was once a HyperCard stack I designed on my Mac Classic. The Complete Triangle and I are very old … er, um … I mean, very old friends.
My fascination with the Complete Triangle as a pedagogical aid is that it encodes a great deal of information about the trigonometric functions, not the least of which is the reason “tangent” is called tangent. (I hate that textbooks define tangent as “sine-over-cosine”, because it strips the word of its meaning and adds to the perception of mathematics as a field of arbitrary buzzwords. But I digress …) The seven similar right triangles that make up the Complete Triangle neatly illustrate many of the fundamental trigonometric identities, such as
$$\sin^2 + \cos^2 = 1 \qquad \text{and} \qquad \frac{\tan}{1}=\frac{\sin}{\cos}$$
(While “tangent is sine-over-cosine” is a lousy definition, it’s a perfectly-good theorem.)
As a dynamic figure with an adjustable angle, the Complete Triangle aids intuition about the overall nature of each individual trig function, such as its range, and relations among them, such as their relative sizes. I could go on and on.
In fact, I did.
With the original release of Trigger, I drafted a 12-page(!) discussion of the Complete Triangle and various trigonometric properties that can be observed within it: “(Almost) Everything You Need to Remember about Trigonometry, in One Simple Diagram”. While I still haven’t gotten around to finalizing the text, my goal is eventually to turn the document into an interactive iBook.
Obviously, I highly recommend working this stuff into any trigonometry curriculum; when I’ve given guest lectures at schools on this topic, students have raved —especially when the lectures came right before exam time— about how the figure is not just a handy mnemonic, but a true encapsulation of much of what they’d been studying, and about how “it all makes sense now”.