I remember teaching integral of sec x to high schoolers with multiplication of sec x + tan x. I mean it is not obvious but it is not like something that would take 100 years.
And the author talks like logarithm was invented long after integration
I know the article is about sec(x) but I want to share this tidbit about its cousin, the hyperbolic secant: sech(x) is its own Fourier transform (modulo rescalings). That’s right, exp(-x^2) is not the only one.
If I understand correctly, the Hermite functions are the eigenfunctions of the Fourier Transform and thus all have this property -- with the Gaussian being a special case. But sech(x) is doubly interesting because it is not a Hermite function, though it can be represented as an infinite series thereof. Are there other well-behaved examples of this, or is sech(x) unique in that regard?
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?
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
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).)
>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.
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.
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.
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.
If we're playing the map-projection-advocacy game, I'd say the Mollweide projection is underrated among equal-area maps [0]. (For local maps, use whatever you want, appropriately centered.) Sure, it distorts shapes away from the central meridian, but locally it only adds a simple horizontal skew. I'm not a big fan of how many equal-area 'compromise' projections lie about how long the lines of latitude are.
>[the Mercator projection] unnecessarily distorts shapes and in particular makes the Americas and Europe look much larger than they actually are. This has been linked, not without rational, to colonialism and racism.
The fact that on many maps Europe is much smaller that it appears should just make you all the more impressed by its achievements.
The most elegant proof IMHO is the one that avoids the original problem entirely.
Int[csc(x) dx] = 2 Int[csc(2u) du]
= 2 Int[du / (2 cos(u) sin(u))]
= Int[sec^2(u) du / tan(u)]
= log(tan(u)) + C
= log(tan(x/2)) + C
Then Int[sec(x)] = Int[csc(u)] = log(tan(u/2)) + C = log(tan(pi/4 - x/2)) + C.
Of course, this was no use to Mercator, because the logarithm hadn't been invented yet. But you aren't just pulling a magic factor out of nowhere. There is definitely a bit of cleverness in rearranging the fraction — you have to be used to trying to find instances of the power rule when dealing with integrals of fractions.
This was the one I was taught in my high school. It has some cleverness (e.g., some trig. transformations) but looks less like coming out of nowhere than the original.
It feels like LLMs could be good contenders for solving symbolically integrals. After spending some time, it really feels like translating between two languages.
Derive 2 for Dos. Green Screen 286 I think or 386 computers in a small side room. Later Windows version was better. Then there was the DOS version of Minitab 5 I think that came as floppy disks in the back of a spiral bound book which I used to generate data sets for students to process for homework so everyone got a slightly different sample.
You can do a lot of numerical maths just with a noddy spreadsheet of course.
A refreshing Hacker News article after a week of repetitive political garbage. Thank you!
I remember teaching integral of sec x to high schoolers with multiplication of sec x + tan x. I mean it is not obvious but it is not like something that would take 100 years.
And the author talks like logarithm was invented long after integration
I know the article is about sec(x) but I want to share this tidbit about its cousin, the hyperbolic secant: sech(x) is its own Fourier transform (modulo rescalings). That’s right, exp(-x^2) is not the only one.
Learned something new today, thank you!
If I understand correctly, the Hermite functions are the eigenfunctions of the Fourier Transform and thus all have this property -- with the Gaussian being a special case. But sech(x) is doubly interesting because it is not a Hermite function, though it can be represented as an infinite series thereof. Are there other well-behaved examples of this, or is sech(x) unique in that regard?
Yes the dirac comb for example. Actually there are infinitely many.
https://en.wikipedia.org/wiki/Dirac_comb
and for other:
http://www.systems.caltech.edu/dsp/ppv/papers/journal08post/...
The impulse train is another well-known one, though I suppose someone will chime in here to rebut that it's arguably not a function.
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?
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
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).)
They were taught to us in Spain, I suppose they don't make an appearance often, but they are perfectly familiar.
In the UK we certainly use sec(x)
speak for your own european country, in my neck of the woods (EE) we were taught and we worked with both secant and cosecant.
I imagine it was more useful when using tables to lookup/approximate the values before calculators with trig support were a thing.
>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.
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.
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.
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?
If we're playing the map-projection-advocacy game, I'd say the Mollweide projection is underrated among equal-area maps [0]. (For local maps, use whatever you want, appropriately centered.) Sure, it distorts shapes away from the central meridian, but locally it only adds a simple horizontal skew. I'm not a big fan of how many equal-area 'compromise' projections lie about how long the lines of latitude are.
[0] https://en.wikipedia.org/wiki/Mollweide_projection
You probably don't live in New Zealand? (Yes I know it's there. Barely.)
>[the Mercator projection] unnecessarily distorts shapes and in particular makes the Americas and Europe look much larger than they actually are. This has been linked, not without rational, to colonialism and racism.
The fact that on many maps Europe is much smaller that it appears should just make you all the more impressed by its achievements.
About how long it'd take me to solve the integral in my calculus finals.
Oh! This was already discussed five years ago with 77 pts and 40 comments (https://news.ycombinator.com/item?id=24304311)
The most elegant proof IMHO is the one that avoids the original problem entirely.
Int[csc(x) dx] = 2 Int[csc(2u) du]
= 2 Int[du / (2 cos(u) sin(u))]
= Int[sec^2(u) du / tan(u)]
= log(tan(u)) + C
= log(tan(x/2)) + C
Then Int[sec(x)] = Int[csc(u)] = log(tan(u/2)) + C = log(tan(pi/4 - x/2)) + C.
Of course, this was no use to Mercator, because the logarithm hadn't been invented yet. But you aren't just pulling a magic factor out of nowhere. There is definitely a bit of cleverness in rearranging the fraction — you have to be used to trying to find instances of the power rule when dealing with integrals of fractions.
This was the one I was taught in my high school. It has some cleverness (e.g., some trig. transformations) but looks less like coming out of nowhere than the original.
It feels like LLMs could be good contenders for solving symbolically integrals. After spending some time, it really feels like translating between two languages.
Wolfram engine was taking integrals just fine way before LLMs were even a thing.
And Lisps too, fitting a sector from a disk:
https://justine.lol/sectorlisp2/
And probably a small forth too, with a dictionary defining every math word, something not so different to Lisp.
LLM's? 4GB of RAM? Your grampa's 486 with 16MB of RAM can do calculus too.
Derive 2 for Dos. Green Screen 286 I think or 386 computers in a small side room. Later Windows version was better. Then there was the DOS version of Minitab 5 I think that came as floppy disks in the back of a spiral bound book which I used to generate data sets for students to process for homework so everyone got a slightly different sample.
You can do a lot of numerical maths just with a noddy spreadsheet of course.
Macsyma, PDP10 + ITS under Maclisp.
https://en.m.wikipedia.org/wiki/PDP-10
https://en.m.wikipedia.org/wiki/Incompatible_Timesharing_Sys...
https://en.m.wikipedia.org/wiki/Macsyma
Fun fact: old Macsyma's math code still runs at is on modern Linux'/BSD's with Maxima. Even plots work the same, albeit in a different output format.
A 386 it's far more powerful than this.