#what the fuck are those #i know the arc versions but what the fuck
A lot of these are more relevant if you're pre-computer and the way you do trig functions is by looking them up in a big careful table. If you're computing 1/sin(x) it saves a step to just look up csc(x)
cot = 1/tan = cos/sin (these 3 save you a division. this will be a pattern, when reading the rest think 'would this save me time if I had to look every trig function up in a table)
sinh, cosh, tanh, csch, sech, coth: in some of your classes you may have used imaginary numbers a+bi. A natural (ish) question is what happens if you do sin(x*i). You end up getting a weird exponential function i*(e^x-e^-x)/2. We take out the imaginary bit and say sinh(x)=i sin(ix). The rest are defined similarly and i like to think of them as the evil mirror universe twins of the trig functions. They're kinda neat if you need to do stuff with hyperbolas.
but mostly, if you have to do sin(a+bi) and you don't have a calculator, you can go "I know the sum of angles formula, I plug in sin(a+bi)=sin(a) cos(bi)+cos(a) sin(bi), and get sin(a) cosh(b) + i cos(a) sinh(b). We got rid of the imaginary numbers in our trig functions yipee, we can look them up in a Big Table now. obviously if we have a calculator we just skip all these steps and never have to use the hyperbolic trig functions.
Arcsin, arccos, and all the other arc- functions undo their respective function.
sinc x = sin(x) / x. In calc you learn that lim x->0 sin(x)/x = 1, and this is very nice and lets you make a lot of simplifying assumptions. When sinc(x) comes up it means that nice assumption isn't good enough and things are going to get very nasty. bessel functions are probably involved. Tanc x = tan(x) / x by the same analogy.
Arcsinh, etc undo their respective hyperbolic function.
cosc x is the only one here i haven't heard of and it's... so the reason sinc works is because lim x->0 sin(x)/x = 1. That isn't true for cosine, cos(0)=1 so if you defined it as cos(x) / x, you'd get divide-by-zero-issues. which is like, fine, but why. I've never seen it. Apparently Julia defines it as the derivative of sinc(x) (NOT equivalent) which adds to the confusion.
sinhc and tanhc are here just to represent their evil mirror twin family. Remember these are all just more compact ways to write i*sin(ix), and honestly more than half the time you'd just see them left in that form.
Arcsinc, arcsinch, etc all undo their respective sinc functions.
Si(x) and Ci(x). hey you know that annoying sinc function? what if you needed to integrate it. Si(x) is the integral from 0 to x of sinc(x) turns out it's impossible. this is a big bummer if you're pre-computer and want your radio to work. good news! we made a chart you can look up the answers on. This comes up a lot in audio engineering where it represents stuff like how long it takes resonances or artifacts to quiet down.
say it with me folks, Shi() and Chi() are here because the evil mirror dimensions twin didn't want to be left out. And again Shi(x) = i Si(ix).
S(x) and C(x) i have genuinely never seen before other than as shorthand for sin and cos? anyone encountered these?
Versine = 1 - cos(x). If you apply Trig Bullshit to this it ends up also being versine = 2 sin^2(x/2).
This is only important because if you want to do any math on a sphere, like, the globe, you use the law of haversines. What's a haversine? it's half a versine. I like haversines because it feels like you're doing pirate math.
Exsecant and excosecant are the versine of secant: exsec(x) = sec(x) - 1. If you don't have a calculator, sec x for small x is something like 1.0004 and so if you're making those aforementioned tables it can be nicer to skip writing the 1.00... and just give the important digits. Again, we have calculators now, they give us enough digits usually.
I will mention here that something they didn't include is Cis(x), or "Cosine plus I Sin". Really this is just e^(i*x) = cos(x) + i sin(x). but a lot of programming languages have a function for it and it's called cis.
is any of this useful, fuck no, but i had fun typing it. aand post 😎
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