# Parabolic trig functions?

**URL:** <https://boards.straightdope.com/t/parabolic-trig-functions/20106>\
**Category:** Factual Questions\
**Created:** [May 21, 2000, 11:00am UTC](https://boards.straightdope.com/t/parabolic-trig-functions/20106 "2000-05-21T11:00:43Z")\
**Posts on this page:** 4\
**Page:** 1

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**Author:** ![bibliophage](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/bibliophage/32/7615_2.png) [@bibliophage](https://boards.straightdope.com/u/bibliophage)\
**Post date:** [May 21, 2000, 11:00am UTC](https://boards.straightdope.com/t/parabolic-trig-functions/20106/1 "2000-05-21T11:00:43Z")

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I found this at the [Texas Instruments site](http://www.ti.com/calc/docs/act/slomer022.htm)

> [@](#):
>
> Consider the equation _y = x[sup]2[/sup] + a sin x_, which defines a family of “parabolic sine” functions.

I know about **hyperbolic** trig functions, but I’ve never heard of a parabolic sine function. Is there some fundamental relationship between this and the circular and hyperbolic functions? The relationship between the circular and hyperbolic functions is quite clear to me. For example, [ul]_i_ sin _u_ = sinh _iu_  
tanh _u_ = -_i_ tan _iu_  
etc.[/ul]Is there a smililarly straightforward relationship that ties in the parabolic functions? Or are these parabolic functions something entirely different? Is it proper to say “sinp x” equals such-and-such?

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**Author:** ![JonF](https://avatars.discourse-cdn.com/v4/letter/j/cab0a1/32.png) [@JonF](https://boards.straightdope.com/u/JonF)\
**Post date:** [May 21, 2000, 11:35am UTC](https://boards.straightdope.com/t/parabolic-trig-functions/20106/2 "2000-05-21T11:35:08Z")

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First I’ve ever heard of them, and tehre sems to be essentially nothing else on the Web …

I think it’s just a term they made up for a function which graphs a little prettier than a parabola or an ordinary sine. The use of the double quotes around the term might indicate this.

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**Author:** ![Zor](https://avatars.discourse-cdn.com/v4/letter/z/848f3c/32.png) [@Zor](https://boards.straightdope.com/u/Zor)\
**Post date:** [May 21, 2000, 4:16pm UTC](https://boards.straightdope.com/t/parabolic-trig-functions/20106/3 "2000-05-21T16:16:38Z")

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Hrm… a parabola with a sine wave superimposed on it. Nothing really special here like **JonF** has already mentioned. Note that the article is supposed to orginate from a high school as teaching material. The contents suggest that this “parabolic sine” was just produced to help students understand the use of derivatives, and how the x[sup]2[/sup] term vanishes while the sine remians etc. etc.

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**Author:** ![Demo](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/demo/32/8472_2.png) [@Demo](https://boards.straightdope.com/u/Demo)\
**Post date:** [May 21, 2000, 8:18pm UTC](https://boards.straightdope.com/t/parabolic-trig-functions/20106/4 "2000-05-21T20:18:38Z")

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It looks like there is a use, actually. I’ve been doing searches and have come across a few sites where it is used to define types of motion (Brownian, for example). The sites have been a bit too advanced for me, as well as using characters that don’t seem to translate very well(maybe need special fonts). Here’s one on [minimal parabolic functions and time-homogeneous parabolic h-transforms](http://www.math.yorku.ca/Who/Faculty/Salisbury/misc.html) and [Molecular Mechanics/Molecular Dynamics](http://www.nap.edu/readingroom/books/mctcc/box2.2.html).

See if you can decipher them… 😉