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]]