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CaptainBoing

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Joined: 07/09/2016
Location: United Kingdom
Posts: 2170

Posted: 09:59am 23 Apr 2022

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Found this in the appendices of some old reference books.

A bit niche but you never now when you might need a Hyperbolic Cosecant. Just stash it away and if you never need it, be happy �

and because I am a sad case, here is the MMBasic for the first group, all tested against Wolfram Alpha. I was going to do the inverse functions too but there is something I need to tweak to get the parsing of the arguments right - it's probably too vague to matter; would anyone ever use them?

Function Sec(x As Float) As Float
Sec=1/Cos(x)
End Function

Function Cosec(x As Float) As Float
Cosec=1/Sin(x)
End Function

Function Cot(x As Float) As Float
Cot=1/Tan(x)
End Function

Function Sinh(x As Float) As Float
Sinh=(Exp(x)-Exp(-x))/2
End Function

Function Cosh(x As Float) As Float
Cosh=(Exp(x)+Exp(-x))/2
End Function

Function Tanh(x As Float)v
Tanh=(Exp(-x)/Exp(x)+Exp(-x))*2+1
End Function

Function Sech(x As Float) As Float
Sech=2/(Exp(x)+Exp(-x))
End Function

Function Cosech(x As Float) As Float
Cosech=2/(Exp(x)-Exp(-x))
End Function

Function Coth(x As Float) As Float
Coth=Exp(-x)/(Exp(x)-Exp(-x))*2+1
End Function
'Wolfram Alpha, for x=2.27. �{-1.55374, 1.30658, -0.840928, 4.78804, 4.89136, 0.978879, 0.204442, 0.208854, 1.02158}

Edited 2022-04-23 20:43 by CaptainBoing

vegipete

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Joined: 29/01/2013
Location: Canada
Posts: 1132

Posted: 04:52pm 23 Apr 2022

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Those first 4 in the second group should be done with ATAN2 instead of ATN.

Visit Vegipete's *Mite Library for cool programs.

CaptainBoing

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Joined: 07/09/2016
Location: United Kingdom
Posts: 2170

Posted: 07:57am 24 Apr 2022

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vegipete said Those first 4 in the second group should be done with ATAN2 instead of ATN.

I did write the inverse functions (second group) into code and they work OK when you respect the input-value ranges, but the odd 2 or 3 don't agree with Wolfram so some remedial work is required.

There is also a typo in one of them (a bracket missing) but can't remember which off the top of my head EDIT: It's Sech_1

For completeness... I would like to get them all working but I don't think I have ever used them and at my stage of life am unlikely to now. It would be a purely mental exercise.

'Inverse trig functions
Function Sin_1(x As Float)'OK
Sin_1=Atn(x/Sqr(-x*x+1))
End Function

Function Cos_1(x As Float)'OK
Cos_1=-Atn(x/Sqr(-x*x+1))+Pi/2
End Function

Function Sec_1(x As Float)' x>1, value different from Wa
Sec_1=Atn(x/Sqr(x*x-1))+Sgn(Sgn(x)-1)*Pi/2
End Function

Function Cosec_1(x As Float)' x>1, OK
Cosec_1=Atn(x/Sqr(x*x-1))+(Sgn(x)-1)*Pi/2
End Function

Function Cot_1(x As Float)' value different from Wa
Cot_1=Atn(x)+Pi/2
End Function

Function Sinh_1(x As Float)' OK
Sinh_1=Log(x+Sqr(x*x+1))
End Function

Function Cosh_1(x As Float)' OK
Cosh_1=Log(x+Sqr(x*x-1))
End Function

Function Tanh_1(x As Float)' OK
Tanh_1=Log((1+x)/(1-x))/2
End Function

Function Sech_1(x As Float) 'value different from Wa
Sech_1=Log(Sqr(-x*x+1)+1)/x
End Function

Function Cosech_1(x As Float)' OK
Cosech_1=Log((Sgn(x)*Sqr(x*x+1)+1)/x)
End Function

Function Coth_1(x As Float)' OK
Coth_1=Log((x+1)/(x-1))/2
End Function

Edited 2022-04-24 18:35 by CaptainBoing

Mixtel90

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Joined: 05/10/2019
Location: United Kingdom
Posts: 7937

Posted: 08:21am 24 Apr 2022

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I'm sure there must be a use for these. Unfortunately my non A-level brain never could get the hang of anything like that. I used Sin, Cos and Tan at school for something - and not for much else since (at least not with any understanding of what I was doing).

They are very impressive though. :)

Mick

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Preliminary MMBasic docs & my PCB designs

CaptainBoing

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Joined: 07/09/2016
Location: United Kingdom
Posts: 2170

Posted: 08:48am 24 Apr 2022

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Yes, I agree these are pretty specialist - I think I used a few of them in my radar work 35+ years ago. TBH I only put them here as it scratches my itch to have all the formulae to maths functions and this reference was the most complete I think I have seen, in some obscure book from the 80s and lost to time... perhaps I breathe on the embers to keep the knowledge easy-access. If it never gets used, no harm done.

As far as school-boy trig goes, the most important thing you need to remember (practically, arguably) is:
Reduce everything to right-angle triangles where you can, then use a2+b2=c2. Got me out of a lot of corners that - all the "base" trig functions will give you everything else from that point.

When I was a lad, I was fearless with this stuff. Now-a-days I still try to apply maths to problems as a first step, taking just 5 or 10 minutes to explore can save a lot of pain later, but it takes a lot of effort. I have forgotten so much!
Edited 2022-04-24 18:49 by CaptainBoing

lizby
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Joined: 17/05/2016
Location: United States
Posts: 3378

Posted: 12:20pm 24 Apr 2022

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Back in 1973-4 I used trig to produce output for automating school bus routing using digital maps. This was basically what Google Maps does, except that the output was to a line printer--many sheets wide and many sheets deep. City of Austin, Texas and surrounding Travis County.

I have no idea at this point what functions I used (SIN, COS?) or how I would now use them.

PicoMite, Armmite F4, SensorKits, MMBasic Hardware, Games, etc. on fruitoftheshed

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