Although the hyperbolic cosine is interesting and I could go on about it.

Eric the Circle for the 18th of June is a bit of geometric wordplay for the week. A secant is — well, many things. One of the important things is it’s a line that cuts across a circle. It intersects the circle in two points. This is as opposed to a tangent, which touch it in one. Or missing it altogether, which I think hasn’t got any special name. “Secant” also appears as one of the six common trig functions out there.

In value the secant of an angle is just the reciprocal of the cosine of that angle. Where the cosine is never smaller than -1 nor larger than 1, the secant is always either greater than 1 or smaller than -1. It’s a useful function to have by name. We can write “the secant of angle θ” as $sec(\theta)$. The otherwise sensible-looking $\cos^{-1}(\theta)$ is unavailable, because we use that to mean “the angle whose cosine is θ”. We need to express that idea, the “arc-cosine” or “inverse cosine”, quite a bit too. And $\cos(\theta)^{-1}$ would look like we wanted the cosine of one divided by θ. Ultimately, we have a lot of ideas we’d like to write down, and only so many convenient quick shorthand ways to write them. And by using secant as its own function we can let the arc-cosine have a convenient shorthand symbol. These symbols are a point where you see the messy, human, evolutionary nature of mathematical symbols at work.

We can understand the cosine of an angle θ by imagining a right triangle with hypotenuse of length 1. Set that so the hypotenuse makes angle θ with respect to the x-axis. Then the opposite leg of that right triangle will be the cosine of θ away from the origin. The secant, now, that works differently. Again here imagine a right triangle, but this time one of the legs has length 1. And that leg is at an angle θ with respect to the x-axis. Then the far leg of that right triangle is going to cross the x-axis. And it’ll do that at a point that’s the secant of θ away from the origin.

Larry Wright’s Motley Classics for the 19th speaks of algebra as the way to explain any sufficiently complicated thing. Algebra’s probably not the right tool to analyze a soap opera, or any drama really. The interactions of characters are probably more a matter for graph theory. That’s the field that studies groups of things and the links between them. Occasionally you’ll see analyses of, say, which characters on some complicated science fiction show spend time with each other and which ones don’t. I’m not aware of any that were done on soap operas proper. I suspect most mathematics-oriented nerds view the soaps as beneath their study. But most soap operas do produce a lot of show to watch, and to summarize; I can’t blame them for taking a smaller, easier-to-summarize data set to study. (Also I’m not sure any of these graphs reveal anything more enlightening than, “Huh, really thought The Doctor met Winston Churchill more often than that”.)

Olivia Jaimes’s Nancy for the 19th is a joke on getting students motivated to do mathematics. Set a problem whose interest people see and they can do wonderful things.

Dave Whamond’s Reality Check for the 19th is our Venn Diagram strip for the week. I say the real punch line is the squirrel’s, though. Properly, yes, the Venn Diagram with the two having nothing in common should still have them overlap in space. There should be a signifier inside that there’s nothing in common, such as the null symbol or an x’d out intersection. But not overlapping at all is so commonly used that it might as well be standard.

Teresa Bullitt’s Frog Applause for the 21st uses a thought balloon full of mathematical symbols as icon for far too much deep thinking to understand. I would like to give my opinion about the meaningfulness of the expressions. But they’re too small for me to make out, and GoComics doesn’t allow for zooming in on their comics anymore. I looks like it’s drawn from some real problem, based on the orderliness of it all. But I have no good reason to believe that.

If you’d like more of these Reading the Comics posts, you can find them in reverse chronological order at this link. If you’re interested in the comics mentioned particularly here, Eric the Circle strips are here. Frog Applause comics are on that link. Motley strips are on that link. Nancy comics are on that page. And And Reality Check strips are here.