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Trigonometric functions under different definition

Real & Complex Analysis, Calculus & Multivariate Calculus, Functional Analysis,
Tolaso J Kos Articles: 2
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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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