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If you define $\cos$ and $\sin$ using their power series, or as the real and imaginary part of the power series of $e^{ix}$, how can you prove that they are periodic? Also, how do you prove that period is $\pi$? And how do you prove that the points $(\cos(x), \ \sin(x))$ for $x \in [0, 2\pi]$ form a circle?

I believe the last question can be proven if you use the continuity of $\cos$ and $\sin$, which follows from their power series definition, and from the fact that $\cos^2(x) + \sin^2(x) = 1$, but using only these two properties is not enough for proving they form a full circle I believe. I think you also need to find their derivatives on the intervals $[0, \ \pi/2]$, $[\pi/2, \ \pi]$, $[\pi, \ 3\pi/2]$ and $[3\pi/2, \ 2\pi]$, is this correct? If so, how can this be done?

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wrt "how do you prove that the points (cos(𝑥), sin(𝑥)) for 𝑥∈[0,2𝜋] form a circle?" see https://gowers.wordpress.com/2014/03/02/how-do-the-power-series-definitions-of-sin-and-cos-relate-to-their-geometrical-interpretations/. wrt earlier questions see the prologue, the exponential function, in Rudin's book Real and Complex Analysis. For an elementary discussion of the exponential function see http://math.mit.edu/~gs/calculus/Article_Exponential.pdf

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