# How do geometric properties of sine and cosine follow from their power series definition?

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?

• The reverse question is asked here: "Deriving the power series for cosine, using basic geometry". You may find my answer interesting.
– Blue
Nov 17, 2018 at 20:25
• @Blue thank you, but I can't quite follow the proof in that answer, especially since the combinatorial argument is omitted. I appreciate the reference and will probably return to it when my math skills are stronger, but for now I'd appreciate a proof in the direction I asked. Nov 17, 2018 at 21:01
• Nov 18, 2018 at 10:44
• Also related: math.stackexchange.com/questions/1048/… May 26, 2021 at 20:16