Talk:Sine and Cosine are Periodic on Reals

have we actually shown anywhere? --Cynic (talk) 02:14, 1 March 2009 (UTC)
 * $$\sin \left({x + \frac \pi 2}\right) = \cos x$$
 * $$\cos \left({x + \frac \pi 2}\right) = -\sin x$$

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. --Matt Westwood 10:17, 1 March 2009 (UTC)