Eikonal Blog

2010.04.09

Simple expression for trigonometric functions of mutiple of an angle

Filed under: mathematics — Tags: , , , , , , — sandokan65 @ 15:16
  • \sin(n x) = \frac{i}2 [(\cos(x)-i \sin(x))^n - (\cos(x)+i \sin(x))^n],
  • \cos(n x) = \frac12 [(\cos(x)-i \sin(x))^n + (\cos(x)+i \sin(x))^n],
  • \tan(n x) = i \frac{(1-i \tan(x))^n - (1+i \tan(x))^n}{(1-i \tan(x))^n + (1+i \tan(x))^n}

Examples:

  • \sin(2x) = 2 \sin(x) \cos(x),
  • \sin(3x) = \sin(x) [3\cos(x)^2-\sin(x)^2],
  • \sin(4x) = 4 \sin(x) \cos(x) [\cos(x)^2-\sin(x)^2],
  • \sin(5x) = \sin(x) [5 \cos(x)^4 - 10 \cos(x)^2 \sin(x)^2 + \sin(x)^4],
  • \cos(2x) = [\cos(x)^2-\sin(x)^2] ,
  • \cos(3x) = \cos(x) [\cos(x)^2- 3 \sin(x)^2] ,
  • \cos(4x) = [\cos(x)^4 - 6 \cos(x)^2 \sin(x)^2 + \sin(x)^4],
  • \cos(5x) = \cos(x) [\cos(x)^4 - 10 \cos(x)^2 \sin(x)^2 + 5 \sin(x)^4],
  • \tan(2x) = \frac{2\tan(x)}{1-\tan(x)^2},
  • \tan(3x) = \frac{\tan(x)[3-\tan(x)^2]}{1-3\tan(x)^2},
  • \tan(4x) = \frac{4\tan(x)[1-\tan(x)^2]}{1-6\tan(x)^2+\tan(x)^4},
  • \tan(5x) = \frac{\tan(x)[5- 10 \tan(x)^2 + \tan(x)^4]}{1-10\tan(x)^2+5\tan(x)^4}, …

Much more on the web:

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