Trig Functions with arguments = Integer * Pi

• To: mathgroup at christensen.cybernetics.net
• Subject: Trig Functions with arguments = Integer * Pi
• From: olness at phyvms.physics.smu.edu (Fredrick Olness (214) 768-2500 or -2495, Fax -4095)
• Date: Wed, 26 Oct 1994 12:36:57 -0500

```> Fedor G. Pikus WRITES:

> I need to declare some variables as integer, so that the trig functions
> would be simplified. I have Declare package from MathSource,
> but it does not help: I declare n to be integer, and Sin[n Pi] stays
> as it is.
=================================================================
Fedor,

I have attached a solution below. I have also included the case of
the Cos[n Pi] which may be of interest.

You may wish to practice on a sample function like sin[] or cos[]
(lowercase) before you start redefining system functions.

Fred Olness
=================================================================
IntegerQ[n] ^= True;
?n

Global`n
IntegerQ[n] ^= True

Unprotect[Sin];
Sin[n_?IntegerQ Pi] := 0;
Protect[Sin];

{Sin[ n Pi],Sin[ 2 n Pi],Sin[ 2 Pi],Sin[ j Pi]}

{0, Sin[2 n Pi], 0, Sin[j Pi]}  (*Note, the 2nd did not simplify*)

Unprotect[Cos];
Cos[n_?IntegerQ Pi] := (-1)^n;
Protect[Cos];

{Cos[ n Pi],Cos[ 2 n Pi],Cos[ 2 Pi],Cos[ 3 Pi]}
n
{(-1) , Cos[2 n Pi], 1, -1}
=================================================================

Fredrick I. Olness

SMU  Mail:  Department of Physics
Fondren Science Bldg.
Southern Methodist University
Dallas, TX 75275

Phone:      (214) 768-2500 (SMU Office)
(214) 768-2495 (SMU Secretary)
(214) 768-4095 (SMU FAX)

Internet:   Olness at phyvms.physics.smu.edu (129.119.200.74)
Olness at mail.physics.smu.edu

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