Talk:Sine and Cosine are Periodic on Reals

have we actually shown

• $\sin \left({x + \frac \pi 2}\right) = \cos x$
• $\cos \left({x + \frac \pi 2}\right) = -\sin x$

anywhere? --Cynic (talk) 02:14, 1 March 2009 (UTC)

Good question. What we did was demonstrate that there is a number $\eta$ such that $\sin$ and $\cos$ are periodic with period $4 \eta$ and then defined $\pi$ as being $2 \eta$. Similarly to how $e$ was defined as being that number such that $\frac d {dx} e^x = e^x$ or exactly how we defined it. What we haven't done is to prove that the ratio of diameter to circumference equals that number which we called $\pi$. Interestingly, I've never come across such a proof. Wherever I've looked, it's either "taken for granted" that the values are the same, or that $\cos$ and $\sin$ are derived geometrically without addressing the analytical approach. So work needs to be done here. --prime mover (talk) 10:17, 1 March 2009 (UTC)