General--Trigonometric functions manipulations in Mathematica
- To: mathgroup at smc.vnet.net
- Subject: [mg71504] General--Trigonometric functions manipulations in Mathematica
- From: ali.abuelmaatti at elec.gla.ac.uk
- Date: Mon, 20 Nov 2006 18:12:07 -0500 (EST)
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]]