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Trying to remember the formula to calculate sin (in radians). I remember it was an infinite sum of increasingly small fractions that would eventually iterate closer and closer to the answer.
It bothers me more that I never understood why the formula worked. If I did, I could just work it out myself (like the quadratic equation).
@me
sin has "odd" symmetry (sin x = -sin -x) which means only odd powers of x in the power series. derivative of sin at x=0 is 1, which means the coefficient of x^1 has to be 1 (the other terms necessarily having zero derivative). Second derivative of sin is -sin. From that you can derive the entire series by writing the first few terms with unknowns for the coefficients, taking the derivative twice, and finding the pattern that tells you each coefficient from the previous one.
@me on a deeper level what's going on is that sin is really exp in disguise and you're just substituting x*sqrt(-1) into the power series for exp.