$$\definecolor{flatred}{RGB}{192, 57, 43} \definecolor{flatblue}{RGB}{41, 128, 185} \definecolor{flatgreen}{RGB}{39, 174, 96} \definecolor{flatpurple}{RGB}{142, 68, 173} \definecolor{flatlightblue}{RGB}{52, 152, 219} \definecolor{flatlightred}{RGB}{231, 76, 60} \definecolor{flatlightgreen}{RGB}{46, 204, 113} \definecolor{flatteal}{RGB}{22, 160, 133} \definecolor{flatpink}{RGB}{243, 104, 224}$$

February 19, 2021

Fundamental Identities (Wrap-Up)

Here's the full set of tangents from the unit circle:

Remember that most references will say that $$\color{flatgreen}\tan(\tfrac\pi2)$$ as undefined — and it is, if you insist that your answer has to be a "real" number. But it's also helpful to think of the answer as $$\color{flatgreen} \overset\sim\infty$$, since this also lets you work with cotangents nicely, as shown in the answers to the Preview Activity. Just be careful — infinity is not a "real" number, so you can't expect it to always behave "nicely" in computations. (That's actually what you'll learn about in calculus — how to properly handle infinity!)

We also started establishing a list of identities — fundamental relationships between the various circular functions. I'll be keeping track of the identities we've learned on a single page to keep them all in one place!

The ones I'd recommend knowing cold are the top three reciprocal identities, the two quotient identities, and the first of the Pythagorean Identities ($${\color{flatblue}\sin^2(\theta)}+{\color{flatred}\cos^2(\theta)} = 1$$). You can always rederive the other ones if you need them (for example by dividing the Pythagorean identities through by $$\color{flatblue}\sin^2(\theta)$$ or $$\color{flatred}\cos^2(\theta)$$, though if you use them enough, you'll find that you'll end up knowing them without having to actively memorize them!