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if we consider a transformation of coordinates

*Cartesian[itex]\rightarrow[/itex]Polar*, it is straightforward to derive [itex]r = (x^2 + y^2)^{1/2}[/itex] and [itex]\theta = atan2(y/x)[/itex], because we actually know what our new coordinate system should be like.

Now let's pretend we have never seen polar coordinates, and we have no idea how to convert from cartesian to polar.

However, all we know is that we have the following family of linear transformations in [itex]\mathcal{R}^2[/itex],

[tex](\begin{array}{ccc} cos\theta & -sin\theta \\ sin\theta & cos\theta \end{array})[/tex]

which are actually rotation matrices acting on 2D vectors, and we would like to find a new frame of reference in which the parameter [itex]\theta[/itex] becomes one

*coordinate*.

Is it possible from this knowledge only, to arrive at the formulas above for [itex]r,\theta[/itex]?