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If a sine curve needs to be oriented at a particular angle, say 45 degrees, we can find the equation of the new curve by converting into Polar coordinates, shifting the angle by 45 degrees and then converting back to Cartesian coordinates. The problem with this is, we end up with equations like (x - y) = sin(x + y) and is difficult to code into a computer program which requires functions to be in the form y = f(x). Is there a way to get the equation in y = f(x) form ?

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If you tilt the sine curve far enough, it won't be a graph of a function; there will be $(x,y)$ and $(x,y')$ on the curve with $y'\ne y$. If you only tilt a little, it will be a graph of a function, but it won't have a nice formula in the shape $y=f(x)$ On the other hand many packages, for instance MAPLE, can plot parametric curves: curves given by $(x(t),y(t))$ for $t$ in a given interval. These won't be fazed by your tilted sine curve, –  Robin Chapman Jul 16 '10 at 10:14
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I believe, although I'm not sure, that you can rotate the sine curve by as much as 45 degrees and still have the resulting curve be the graph of a function. (This is because the derivative of the sine is between 1 and -1; 45 degrees is the arctangent of 1.) –  Michael Lugo Jul 16 '10 at 17:02
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