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copied!<p>Using one of the most beautiful formulas of mathematics, <a href="http://en.wikipedia.org/wiki/Euler%27s_formula" rel="nofollow noreferrer">Euler's formula</a><br> <code>exp(i*x) = cos(x) + i*sin(x)</code>,</p> <p>substituting <code>x := n * phi</code>:</p> <p><code>cos(n*phi) = Re( exp(i*n*phi) )</code><br> <code>sin(n*phi) = Im( exp(i*n*phi) )</code></p> <p><code>exp(i*n*phi) = exp(i*phi) ^ n</code></p> <p>Power <code>^n</code> is <code>n</code> repeated multiplications. Therefore you can calculate <code>cos(n*phi)</code> and simultaneously <code>sin(n*phi)</code> by repeated complex multiplication by <code>exp(i*phi)</code> starting with <code>(1+i*0)</code>.</p> <p>Code examples:</p> <p>Python:</p> <pre><code>from math import * DEG2RAD = pi/180.0 # conversion factor degrees --> radians phi = 10*DEG2RAD # constant e.g. 10 degrees c = cos(phi)+1j*sin(phi) # = exp(1j*phi) h=1+0j for i in range(1,10): h = h*c print "%d %8.3f"%(i,h.real) </code></pre> <p>or C:</p> <pre><code>#include <stdio.h> #include <math.h> // numer of values to calculate: #define N 10 // conversion factor degrees --> radians: #define DEG2RAD (3.14159265/180.0) // e.g. constant is 10 degrees: #define PHI (10*DEG2RAD) typedef struct { double re,im; } complex_t; int main(int argc, char **argv) { complex_t c; complex_t h[N]; int index; c.re=cos(PHI); c.im=sin(PHI); h[0].re=1.0; h[0].im=0.0; for(index=1; index<N; index++) { // complex multiplication h[index] = h[index-1] * c; h[index].re=h[index-1].re*c.re - h[index-1].im*c.im; h[index].im=h[index-1].re*c.im + h[index-1].im*c.re; printf("%d: %8.3f\n",index,h[index].re); } } </code></pre>

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