       Re: Trig Simplifications

• To: mathgroup at smc.vnet.net
• Subject: [mg3710] Re: Trig Simplifications
• From: Harald Berndt <haraldb at nature.berkeley.edu>
• Date: Wed, 10 Apr 1996 02:11:54 -0400
• Organization: University of California Forest Prodcts Lab
• Sender: owner-wri-mathgroup at wolfram.com

```Alexander Casti wrote:
>
> Basically my question is how to get mathematica to write
>
> Exp[I x] in place of Cos[x] + I Sin[x]
>
> Suppose I have the expression
>
> In: f[x_]= Cos[x] + I Sin[x] - Cos[2 X] + I Sin[2 X]
>
> I would like mathematica to simplify this into
>
> f[x_]= Exp[I X] - Exp[2 I X]
>
> I tried the substitution command
>
> In: % ./ Cos[X]+I Sin[X]-> Exp[I X] and it returned
> Out: E^{I X}  - Cos[2 X] + I Sin[2 X]
>
I believe you don't really want what you've written out, because:
Cos[x] + I Sin[x] - Cos[2 X] + I Sin[2 X] == Exp[I X] - Exp[- 2 I X].

That sign error is one part of the problem: one needs two replacement
rules,

Cos[t_]+I Sin[t_]-> Exp[I t]

and

Cos[t_]-I Sin[t_]-> Exp[-I t].

Using Pattern[t, Blank[]] (the t_) will take care of the different
variable names x and 2 X, but there are still two substitutions
required. One needs to use ReplaceRepeated[] (//.).

Cos[x] + I Sin[x] + Cos[2 X] + I Sin[2 X]//.Cos[t_]+I Sin[t_]-> Exp[
I t]

will work and so will

Cos[x] + I Sin[x] - Cos[2 X] + I Sin[2 X]//.{Cos[t__]+I Sin[t__]->
Exp[I t], -(Cos[t_]-I Sin[t_])-> -Exp[- I t]}

This still seems clumsy and I hope someone will suggest a more
efficient way of writing this.
--
______________________________________________________________________
Harald Berndt,                                University of California
Research Specialist                         Forest Products Laboratory

Phone:	510-215-4224                                FAX:			510-215-4299

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```

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