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Taylor Series of Trig Functions

raiyanshadow Dec 30th, 2019 235 Never
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  1. The $sin(x)$, $cos(x)$ and $tan(x)$ can be founded using the Maclaurian/Taylor series, demonstrated below:
  2.  
  3. $$sin(x) = x - \!\frac {x^3}{3!}\ + \!\frac {x^5}{5!}\ - \!\frac {x^7}{7!}\ + \cdot\cdot\cdot $$
  4. $$=\sum_{n=0}^{\infty}\!\frac {{(-1)^n}{x^{2n+1}}}{(2n+1)!}, $$
  5. $$cos(x) = 1 - \!\frac {x^2}{2!}\ + \!\frac {x^4}{4!}\ - \!\frac {x^6}{6!}\ + \cdot\cdot\cdot$$
  6. $$= \sum_{n=0}^{\infty}\!\frac {(-1)^n{x^{2n+1}}}{(2n)!},$$
  7. Both the $sin(x)$ and $cos(x)$ will apply for all $x \in {\rm I\!R}$  .
  8. $$tan(x) = x + \!\frac {1}{3}x^3\ + \!\frac {2}{15}x^5\ + \!\frac {17}{315}x^7\ +\cdot\cdot\cdot$$
  9. $$=\sum_{n=1}^{\infty}\!\frac {(-1)^{n-1}2^{2n}(2^{2n}-1)B_{2n}x^{2n-1}}{(2n)!}$$
  10. $$=\sum_{n=0}^{\infty}\!\frac {U_{2n+1}x^{2n+1}}{(2n+1)!}$$
  11. The $tan(x)$ will only apply for $|x| < \!\frac {\pi}{2}$
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