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Re: Avoid the use of certain functions

  • To: mathgroup at smc.vnet.net
  • Subject: [mg111108] Re: Avoid the use of certain functions
  • From: Murray Eisenberg <murray at math.umass.edu>
  • Date: Tue, 20 Jul 2010 03:45:30 -0400 (EDT)

One should not be so quick to condemn the world when it does not conform 
to your own preconceptions and preferences!

There is a good geometric reason that sec is an appropriate name for the 
reciprocal of cos rather than of sin.  Draw the unit circle, center at 
the origin O. Mark off a central angle of theta from the positive 
x-axis, and let P be the corresponding point on the circle. Draw the 
vertical line L through the point (1,0). Extend the ray OP until it 
intersects L at a point Q. Then the length of OQ is sec (theta).  That 
line segment OQ cuts across the circle.

[See, e.g., http://en.wikipedia.org/wiki/Trigonometric_functions. (I 
thought I remembered also some Wolfram Demonstration of this, but I 
cannot locate it now.)]

Furthermore, in the usual list of the 6 trig functions

   sin
   cos
   tan
   cot
   sec
   csc

there is a symmetry of pairs around the middle: cot = 1/tan, sec = 
1/cos, csc = 1/sin.

Finally, what's wrong with sec and csc?  Mathematical expressions are 
often simpler when expressed in terms of them, since this avoids 
(explicit) use of fractions for the functions.

On 7/19/2010 2:10 AM, AES wrote:
>> From: Sam Takoy [mailto:sam.takoy at yahoo.com]
>>
>> Hi,
>>
>> Is there a way to ask Mathematica to avoid expressing answers in terms
>> of certain functions. For example, I [can't?] stand Sec, Csc, Sech, and Csch
>> and would rather see Sec^-1, etc.
>
> I'm with you on this one:  always hated Sec and Csc (and never
> understood the backward naming of these functions -- why isn't Sec =
> 1/Sin and Csc = 1/Cos?)
>
> So, I'd like the various Simplifiy and XxxToYyy functions in Mathematica to
> always give precedence to Sin and Cos by default, and avoid using Sec
> and Csc whenever possible, even if this produces some "1 overs" in the
> output expression.
>
> But I suspect trying to implement this at this point would require more
> complexity than it would be worth.
>

-- 
Murray Eisenberg                     murray at math.umass.edu
Mathematics & Statistics Dept.
Lederle Graduate Research Tower      phone 413 549-1020 (H)
University of Massachusetts                413 545-2859 (W)
710 North Pleasant Street            fax   413 545-1801
Amherst, MA 01003-9305


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