       Semi-symbolically Semi-numerically

• To: mathgroup at smc.vnet.net
• Subject: [mg113387] Semi-symbolically Semi-numerically
• From: Sam Takoy <sam.takoy at yahoo.com>
• Date: Tue, 26 Oct 2010 05:35:30 -0400 (EDT)

Hi,

I'm working on a project where certain elements need to be computed
numerically. Not surprisingly, some simplifications that work
symbolically, don't work with floating numbers. But some still do, e.g.
1.0 Sin[a]^2 + 1.0 Cos[a]^2 // Simplify is 1.0;

Is there a way to make the example that follows, that starts with Cos[8
theta] then goes to Cartesian coordinates and back to polar, to work as
well as the trivial trig example above?

Many thanks in advance,

Sam

toCart = theta -> ArcTan[x, y];
toPolar = {x -> Cos[theta], y -> Sin[theta]};
help = {Cos[ArcTan[x, y]] -> x/Sqrt[x^2 + y^2],
Sin[ArcTan[x, y]] -> y/Sqrt[x^2 + y^2]};

((Cos[8 theta] /. toCart // TrigExpand) /. help // FullSimplify) /.
toPolar // Simplify
((1.0 Cos[8 theta] /. toCart // TrigExpand) /. help //
FullSimplify) /. toPolar // Simplify

Out= Cos[8 theta]

Out=
1. Cos[theta]^8 - 28. Cos[theta]^6 Sin[theta]^2 -
28. Cos[theta]^2 Sin[theta]^6 + 1. Sin[theta]^8 +
4.375 Sin[2 theta]^4

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