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Comment by cl3misch | Hacker News Reader

Neither 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?

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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..

See R.W. Sinnott, "Virtues of the Haversine", Sky and Telescope, vol. 68, no. 2, 1984, p. 159

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 imagine it was more useful when using tables to lookup/approximate the values before calculators with trig support were a thing.

speak for your own european country, in my neck of the woods (EE) we were taught and we worked with both secant and cosecant.

In the UK we certainly use sec(x)

They were taught to us in Spain, I suppose they don't make an appearance often, but they are perfectly familiar.

there are these old-fashioned looking drawings...

(quick search, didn't find the old ones, but similar to these)

https://mathematicaldaily.weebly.com/secant-cosecant-cotange...

https://www.pinterest.com/pin/enter-image-description-here--...

... which were not used in my education but whenever i saw them i wished they had been, they lay out a geometric interpretation of all of them. by "old" i mean "look like Leonardo drew them"

the only other one is cot, actually.

Personally I thought they were nice to have because coming up with the integral of 1/cos on the fly is pretty brutal in a long integral

I used sec, cosec and others during my math degree in the UK too.

I'm sure you used inverse of a cosine multiple times. Didactic math today is just not bothering to give it a name. Probably because people think that sin, cos and tan is enough. Even ctg which is just inverse of tan is often skipped.

The secant is the reciprocal of a cosine – the hypotenuse over the adjacent

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?

The weird thing about 1/cos is it’s discontinuous wherever cos is 0 but, yes, it’s a function.

Yeah, that was replied to the wrong comment.

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

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 x^{-1} that means 1/x. If you write f^{-1} and f is a function, then _generally_ it means the inverse of f. In the case of trig functions this is doubly confusing because people write sin^2 theta meaning (sin theta)^2 but sin^-1 theta means arcsin theta.

That's why in my maths studies they started by teaching you to do the inverse with a -1 so when you see it you don't get confused but changed to preferring arcsin etc as this is unambiguous and if you learn to write this way you won't confuse others.

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.

Inverse function: https://en.wikipedia.org/wiki/Inverse_function / https://fr.wikipedia.org/wiki/Bijection_r%C3%A9ciproque

Reciprocal: https://en.wikipedia.org/wiki/Multiplicative_inverse / https://fr.wikipedia.org/wiki/Inverse

Wikipedia seems to have chosen "multiplicative inverse" over "reciprocal" for title, even though they are clearly indicated as synonymous.

That’s a really good point. I will try to remember to do that in future.

>Neither in (German) high school nor in the many math courses of a physics B.Sc. have I ever used the secant function

I think we used it in geometry in US high school, but only to complete an assignment or two to show we could use trig functions correctly. I had to relearn how all of them worked to help my kid with homework, it's mostly look at the angles and sides you have available and pick which trig function is necessary to figure out which one you're solving for. I'm sure there are real life uses for trig functions, and I hate to be one of those "when are we ever going to use this" types, but I've never used any of them outside of math classes.

Law of sines is useful when constructing things with a known angle outside of CAD.

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