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1. e^x, sin(x) and cos(x) can each be expanded out into an infinite sequence of fractions (Maclaurin series), each fraction of the form x^n / factorial(n). This relies on differential calculus, Taylor series, theory of limits, convergence etc. 2. The e series fractions contain all the integers in the numerator (x^0, x^1, x^2, etc), while the sin series has only odd integers and the cosine series has only even integers. Also, e series terms are all additive while the trig functions series alternate adding and subtracting each successive fraction. 3. Introducing complex numbers (i = square root of negative one), we can generate the series for e^ix, which can be shown to be equal to sin(x) + i * cos(x). Note that introducing i into the e series means we generate a negative term for the even fractions in the e series (squaring i gives us -1), which is why i is so necessary here. 4. Solving e^ix for x = pi, using sin(x) + icos(x), we get -1. Mathologer: https://www.youtube.com/watch?v=-dhHrg-KbJ0 and https://www.youtube.com/watch?v=DoAbA6rXrwA As far as why an exponential function like e^x should have anything fundamental linking it to trigonometric functions like sin(x) and cos(x), it is rather strange. |