I know that many mathematic methods have been used for many millenniums but it is not necessarily correct without proper foundation. To say that sin(2Πradians) is defined circularly would be a bit disingenuous of me. I do wonder why it is chosen that the axis of reflection is centered on the center of the circle. The equation is x^{2} + y^{2} = C ; as one example, however it can be defined in other ways. I suppose I could say (x-j)^{2} + (y-k)^{2} = C. If the zero were left and below the circle, there would be no negative sines or cosines.

In a rather round about way ( pun intended ) they have stumbled into the quadrant issue of dimensionality and merged quantity with topology. If I define a program that does an arithmetic series and I modify the methods there to determine what outcome is achieved for different perspectives, the results can often be the same, without implicit topology or vectors. It seems to me that in a logical sense they are simply projecting results on a circle or sphere and then wondering why it seems so convoluted.

In terms of degrees of rotation about a circle, it seems rather arbitrary, though necessary to have a standard. In the computer it is a different issue as I can just switch the scales and the computer never gets confused by a new standard, as it only knows what I tell it.

A function is defined as having a single 'y' for every 'x' and so a circle does not represent a function and projecting solutions on circles or any other form is an advantage. I am going to consider if there might be a way to untangle all of this into a single rule set that can be applied in my program and measure results to determine if it is more predictive, or usable or faster and if I can't resolve a new method in some reasonable time, I will punt and go with the way it is defined currently, and attack that windmill again another day.

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