Re: Sin Cos Tan, why not Sin Sec Tan?

From: Cassandra Thompson (cass.harley_at_bigpond.com)
Date: 07/15/04 

Date: Thu, 15 Jul 2004 06:59:38 GMT

Jake Wildstrom wrote:

> The Prophet Cassandra Thompson known to the wise as cass.harley@bigpond.com, opened the Book of Words, and read unto the people:
>
>>Why is it important that the hypotonuse be the denomitator? Because the
>>answer will always be <=1. Because as far as I can tell that is the only
>>reason for choosing the sin/cos pair over the sec/csc pair.
>>
>>I am guessing that there is a very good answer why this is better then
>>the result being >1, however I cannot yet see why.
>
>
> Well, bounded functions are frequently nicer. In calculus and
> analysis, you'll find there are many things which sin and cos
> "naturally" fall out of (for instance, there's a grand cosmic
> relationship between the exponential function e^x and the sin and cos
> functions; also, the sin and cos functions, unlike the sec and csc
> functions, have good approximating sequences and behave generally
> nicely in ways which sec and csc fail to do. One last, simpler,
> example: sin and cos are defined everywhere; sec and csc have
> discontinuities when they jump to infinity, which is really unpleasant
> (tan has this problem too, but with the tan/cot pair, this is
> unavoidable; the distinction of one of tan/cot as 'customary' is
> moderately arbitrary, but the distinction of sin/cos as customary and
> sec/csc as exotic is in fact a nonarbitrary decision).
>
> +-------------------------------------------------------------+
> | D. Jacob Wildstrom -- Math monkey and freelance thinker |
> | Graduate Student, University of California at San Diego |
> | "A mathematician is a device for turning coffee into |
> | theorems." -Alfred Renyi |
> +-------------------------------------------------------------+
>
> The opinions expressed herein are not necessarily endorsed by the
> University of California or math department thereof.
Thank you (and everybody else who has responded). You have really
clarified what I was trying to understand, and I feel like I have a much
stronger knowledge about it.

Thanks again.
Cassandra Thompson.