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Trigonometric functions under different definition
https://www.mathimatikoi.org/forum/viewtopic.php?f=3&t=942
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Author:  Tolaso J Kos [ Thu Jul 14, 2016 1:03 pm ]
Post subject:  Trigonometric functions under different definition

An other way to define the trigonometric functions is by using their power series, that is: $$ \sin x = \sum_{n=0}^{\infty}\frac{(-1)^n x^{2n+1}}{\left ( 2n+1 \right )!},\; \cos x = \sum_{n=0}^{\infty}\frac{(-1)^n x^{2n}}{\left ( 2n \right )!}, \; x \in \mathbb{R} $$ The classic definition of the trigonometric functions is based on the unit circle.

a. Use the definition given above to prove that:
  1. \( \sin 0 =0, \; \cos 0 =1 \)
  2. \( \left ( \sin x \right )'= \cos x , \; \left ( \cos x \right )' =-\sin x \)
  3. \( \sin^2 x + \cos^2 x =1 \)

b. Prove that the classic definition, the definition given above and the definition \( \displaystyle \sin x = \frac{e^{ix}-e^{-ix}}{2i}, \; \cos x =\frac{e^{ix}+e^{-ix}}{2} \) are equivalent.

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