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[&#937;1] + A Sin[&#937;2]

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[&#937;1] + A2Sin[&#937;1] 2 + A3 Sin[&#937;1]3 + A Sin[&#937;2] + 2A2 Sin[&#937;1] Sin[&#937;2] +  3A3 Sin[&#937;1]2 Sin[&#937;2] + A2 Sin[&#937;2]2 + 3A3 Sin[&#937;1] Sin[&#937;2]2 +  A3 Sin[&#937;2]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[&#937;1] + Sin[&#937;2]) (1 + A2 Sin[&#937;1]2 + A Sin[&#937;2] + A2 Sin[&#937;2]2 + A Sin[&#937;1] (1 + 2A Sin[&#937;2]))

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[&#937;1+ &#937;2]=Sin[&#937;1] Cos[&#937;2] + Cos[&#937;1] Sin[&#937;2],

Sin[&#937;1- &#937;2]=Sin[&#937;1] Cos[&#937;2] - Cos[&#937;1] Sin[&#937;2],

Cos[&#937;1+ &#937;2]= Cos[&#937;1] Cos[&#937;2] - Sin[&#937;1] Sin[&#937;2],

Cos[&#937;1- &#937;2]= Cos[&#937;1] Cos[&#937;2] + Sin[&#937;1] Sin[&#937;2],

Cos[&#937;1]2=.5 + .5 Cos[2&#937;1]

And

Cos[&#937;1]2=.5 - .5 Cos[2&#937;1]

To produce the terms that has [&#937;1+ &#937;2], [&#937;1+ &#937;2], [2&#937;1] or [2&#937;2]. 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 [&#937;1+ &#937;2], [&#937;1+ &#937;2], [2&#937;1] or [2&#937;2] terms?

Thanks all in advance for reading and for your help.

Ali

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```

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