Matrix of arguments and amplitudes of cosinus with software (MatLab, Maple, Mathematica...)

Question

My problem is that, I have a big matrix with $6\times 6$ elements, with each term of the matrix is a big function of sum of cosines where I need to work only with the arguments and the amplitudes separately.

Simplifying the task is, I have a function like:

$$f(\phi) = A_{1} \cos \left(\phi -\frac{2 \pi }{3}\right)+ A_{2} \cos \left(2 \phi +\frac{\pi }{3}\right)+ A_{3} \cos \left(3 \phi -\frac{\pi }{2}\right);$$

and with any software like Matlab, Maple, Mathematica or any software. Input the matrix $f(\phi)$ and have the output a vector of amplitudes and a vector of arguments:

$$A=\left[ \begin{array}{ccc} A1 & A2 & A3 \end{array} \right] \text{and} \arg= \left[ \begin{array}{ccc} \phi -\frac{2 \pi }{3} & 2 \phi +\frac{\pi }{3} & 3 \phi -\frac{\pi }{2} \end{array} \right]$$

I have tried with Regexp of MatLab but it is not possible to choose only the argument of the cosine.

If anyone can help me please, Thank you very much

Answer 1

Mathematica handles automatically provides these reductions $$ \begin{align} % \cos \left(\phi -\frac{2 \pi }{3}\right) &= -\sin \left(\frac{\pi }{6}-\phi \right) \\ % \cos \left(2 \phi +\frac{\pi }{3}\right) &= \sin \left(\frac{\pi }{6}-2 \phi \right) \\ % \cos \left(3 \phi -\frac{\pi }{2}\right) &= \sin (3 \phi ) % \end{align} $$

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The TrigReduce expression will yield $$ \text{TrigReduce}\left[A_{1} \cos \left(\phi -\frac{2 \pi }{3}\right)+A_{2} \cos \left(2 \phi +\frac{\pi }{3}\right)+A_{3} \cos \left(3 \phi -\frac{\pi }{2}\right)\right] = -A_{1} \sin \left(\frac{\pi }{6}-\phi \right)+A_{2} \sin \left(\frac{\pi }{6}-2 \phi \right)+A_{3} \sin (3 \phi ) $$

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Using TrigExpand $$ \text{TrigExpand}\left[A_{1} \cos \left(\phi -\frac{2 \pi }{3}\right)+A_{2} \cos \left(2 \phi +\frac{\pi }{3}\right)+A_{3} \cos \left(3 \phi -\frac{\pi }{2}\right)\right] = \frac{1}{2} \sqrt{3} A_{1} \sin \phi -\frac{1}{2} A_{1} \cos \phi -\frac{1}{2} A_{2} \sin ^2(\phi )+\frac{1}{2} A_{2} \cos^{2}\phi -\sqrt{3} A_{2} \sin \phi \cos \phi -A_{3} \sin ^3(\phi )+3 A_{3} \sin \phi \cos^{2}\phi $$

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Using Simplify or FullSimplify: $$ \text{Simplify}\left[A_{1} \cos \left(\phi -\frac{2 \pi }{3}\right)+A_{2} \cos \left(2 \phi +\frac{\pi }{3}\right)+A_{3} \cos \left(3 \phi -\frac{\pi }{2}\right)\right] = -A_{1} \sin \left(\frac{1}{6} (\pi -6 \phi )\right)+A_{2} \cos \left(\frac{1}{3} (6 \phi +\pi )\right)+A_{3} \sin (3 \phi ) $$

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TrigFactor $$ \text{TrigFactor}\left[A_{1} \cos \left(\phi -\frac{2 \pi }{3}\right)+A_{2} \cos \left(2 \phi +\frac{\pi }{3}\right)+A_{3} \cos \left(3 \phi -\frac{\pi }{2}\right)\right] = -\frac{1}{2} \sqrt[3]{-1} \left(\left((-1)^{5/6} A_{1}+i A_{1}\right) \sin (\phi )+\left(A_{1}-\sqrt[3]{-1} A_{1}\right) \cos (\phi )+\left(-(-1)^{5/6} A_{2}-i A_{2}\right) \sin (2 \phi )+\left(\sqrt[3]{-1} A_{2}-A_{2}\right) \cos (2 \phi )+2 (-1)^{2/3} A_{3} \sin (3 \phi )\right) $$

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The trick is to decide which of these forms to work with and use FullForm. For example $$ \text{FullForm}\left[A_{1} Cos\left[\phi - \frac{2}{3} \pi\right]\right] = \text{Times}[\underbrace{A_{1}}_{amplitude},\text{Cos}[\underbrace{\text{Plus}[\phi ,\text{Times}[-1,\text{Times}[\text{Times}[2,\text{Power}[3,-1]],\text{Pi}]]]}_{angle}]] $$ Extract

This problem seems better suited for the crowd at [Mathematica Stack Exchange][2]

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Solution