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Re: General--Trigonometric functions manipulations in Mathematica
*To*: mathgroup at smc.vnet.net
*Subject*: [mg71512] Re: General--Trigonometric functions manipulations in Mathematica
*From*: "dimitris" <dimmechan at yahoo.com>
*Date*: Tue, 21 Nov 2006 07:05:09 -0500 (EST)
*References*: <ejtdp7$j4f$1@smc.vnet.net>
Hello.
It's not clear to me all the content of your post, so I concentrate
only to your question.
But first two things to notice.
Don't use special characters like greek capital omega in your posts.
You see how nasty they were appeared?
I was able to see the variables names after I read MathGroup mailing
list!
Also avoid to mix exact and inexact numbers; e.g. don't mix 2 with 0.5.
Head /@ {2, 2.}
{Integer, Real}
InexactNumberQ /@ {2., N[Pi], Pi, E, N[E], 2 - 6.*I, 5/4, Gamma}
{True, True, False, False, True, True, False, False}
Your question now.
For the identities like Sin[a + b] = Sin[a] Cos[b] + Cos[a] Sin[b] you
use TrigExpand
TrigExpand /@ {Sin[a+ b], Sin[a - b], Cos[a + b], Cos[a-
b]}//ColumnForm
FrontEndExecute[{HelpBrowserLookup["MainBook", "3.3.7"]}]
For the identities 2Cos[a] =1/2+ 1/2Cos[2a] and Sin[a]= 1/2 -
1/2Cos[2a] you can define your own rules as follows:
trigrule1 = 2*Cos[x_] :> 1/2 + (1/2)*Cos[2*x];
trigrule2 = 2*Sin[x_] :> 1/2 - (1/2)*Cos[2*x];
Then
2*Cos[a] /. trigrule1
2*Sin[a] /. trigrule2
1/2 + (1/2)*Cos[2*a]
1/2 - (1/2)*Cos[2*a]
Regards
Dimitris
ali.abuelmaatti at elec.gla.ac.uk wrote:
> Hi all,
>
> I am trying to perform the following operation:
>
> Let us assume I have a two-tone input to my system in the form
>
> x = A Sin[Ω1] + A Sin[Ω2]
>
> and our system is nonlinear so it looks like this
>
> y = x + x2 + x3
>
> Now to substitute x (the input) into y (the system) I use the expand function as follows
>
> Expand [y]
>
> Out[1]= ASin[Ω1] + A2Sin[Ω1] 2 + A3 Sin[Ω1]3 + A Sin[Ω2] + 2A2 Sin[Ω1] Sin[Ω2] + 3A3 Sin[Ω1]2 Sin[Ω2] + A2 Sin[Ω2]2 + 3A3 Sin[Ω1] Sin[Ω2]2 + A3 Sin[Ω2]3
>
> Now that is good as it is substituted properly but this is just direct substitution and it is not what I am looking for, I want to break all the high power terms on the Sin functions so I try using simplify as follows:
>
> Simplify[Out[1]]
>
> Out[2]= A (Sin[Ω1] + Sin[Ω2]) (1 + A2 Sin[Ω1]2 + A Sin[Ω2] + A2 Sin[Ω2]2 + A Sin[Ω1] (1 + 2A Sin[Ω2]))
>
> Now it looks better but unfortunately it doesnÂ?t get a lot further than that.
>
> The question is, how can I persuade Mathematica to use some of the well known trigonometric functions for example:
>
> Sin[Ω1+ Ω2]=Sin[Ω1] Cos[Ω2] + Cos[Ω1] Sin[Ω2],
>
> Sin[Ω1- Ω2]=Sin[Ω1] Cos[Ω2] - Cos[Ω1] Sin[Ω2],
>
> Cos[Ω1+ Ω2]= Cos[Ω1] Cos[Ω2] - Sin[Ω1] Sin[Ω2],
>
> Cos[Ω1- Ω2]= Cos[Ω1] Cos[Ω2] + Sin[Ω1] Sin[Ω2],
>
> Cos[Ω1]2=.5 + .5 Cos[2Ω1]
>
> And
>
> Cos[Ω1]2=.5 - .5 Cos[2Ω1]
>
> To produce the terms that has [Ω1+ Ω2], [Ω1+ Ω2], [2Ω1] or [2Ω2]. These are the terms that I am most interested it which do come out of that non linear system when fed with a two-tone input.
>
> Which functions can I use to give me that, give me these Sin or Cos [Ω1+ Ω2], [Ω1+ Ω2], [2Ω1] or [2Ω2] terms?
>
> Thanks all in advance for reading and for your help.
>
> Ali
>
>
> Link to the forum page for this post:
> http://www.mathematica-users.org/webMathematica/wiki/wiki.jsp?pageName=Special:Forum_ViewTopic&pid=15508#p15508
> Posted through http://www.mathematica-users.org [[postId=15508]]
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