This is also the inverse Gudermannian function [1]. That Wikipedia page has some nice geometrical insights.
100 Years to Solve an Integral (2020)
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Re: 100 Years to Solve an Integral (2020)
#62Earlier quoted context omitted.
Trig is full of functions that fall into disuse and are forgotten. For example "versine" versin theta = 1-cos theta. There is also "haversine" which is (1-cos theta)/2. Which is used in navigation apparently https://en.wikipedia.org/wiki/Versine
iirc, haversine is useful for transforming 2-d "as the crow flies" coords to their 3-d equivalents. at longer distances a body's curvature is really noticeable and often overlooked
Interesting. versine has a lovely and intuitive geometric definition. If you construct a right triangle from the origin to some point on the circle, most people who have done trig will know that the x-coordinate of that point is r cos theta, where theta is the angle and r is the radius. Geometrically the distance from the origin to where the triangle rests on the x axis is r cos theta. But what about the rest of that radius? ie the line segment on the x-axis from there to where the circle intersects the x-axis?
That is r versin theta (ie r - r cos theta). Pretty cool no? I mean I've literally never had to find the length of that line, but that's how you would if you wanted to..
Re: 100 Years to Solve an Integral (2020)
#63Earlier quoted context omitted.
I know what you mean, but as a sibling pointed out for everyone else's benefit, parent is using the word inverse where they mean reciprocal. The inverse of cosine is arccosine (sometimes written acos or cos^{-1}). Secant is the reciprocal of cos ie sec x = 1/cos(x)). Likewise cotan is the reciprocal of tan (1/tan). The inverse of tan is atan/arctan/tan^{-1}. This is confusing for a lot of people because if you write…
It does not help that both reciprocal and inverse come from French, and that their common meanings are reversed in English. I'm not sure whether the meaning of both words has remained constant over time in these two languages, as they both roughly mean "the opposite" and if you want to avoid ambiguity, you simply add context. For example, if you say "inverse function" or "multiplicative inverse" it's not ambiguous. I…
That’s a really good point. I will try to remember to do that in future.
Re: 100 Years to Solve an Integral (2020)
#64Earlier quoted context omitted.
That’s right, it’s a distribution. And that fact has me, a non-mathematician, personally caused some huge headaches, because I thought I could treat it just like a function… Yeah, turns out really weird things happen if you try to do so without knowing what you’re doing. For example, taking its square does not make sense.
It is a function. What do you mean?
Oops, replied to the wrong comment. This is the one I meant to reply to, which is talking about the impulse train, which is not a function: https://news.ycombinator.com/item?id=43741539
Re: 100 Years to Solve an Integral (2020)
#65Re: 100 Years to Solve an Integral (2020)
#66Neither in (German) high school nor in the many math courses of a physics B.Sc. have I ever used the secant function. I am surprised the article does not explain it in the beginning. I assume for other people it must be a common function?
It's a US thing. Europeans just write 1/cos(x) instead of treating it as a special thing with its own name. The Americans have sec, csc, and a bunch of others I never bothered to learn. It doesn't seem to add all that much to me? (Of course, it's a bit hypocritical since I gladly use tan(x).)
I used sec, cosec and others during my math degree in the UK too.