Two basic questions on Nyquist-Shannon Theorem for sinusoide

Question

Consider the following sinusoid signal with frequency $F_0$: $$ x(t)=a\sin(2\pi F_0t) $$ If we sample $x(t)$ every $T$ seconds, we get the following discrete signal: $$ x[nT]=a\sin(2\pi F_0nT) $$

(a) Does Nyquist–Shannon sampling theorem claims that if $T<\frac{1}{2F_0}$, then $x(t)$ is uniquely determined by $X[nT]$? How do I state that Theorem (mathematically)?

(b) How to prove the theorem only for sinusoids? I noticed that when sampling $x_k(t)=a\sin(2\pi(F_0+\frac{k}{T}))$ we get the same discrete signal. How to proceed?

Answer 1

It is a corollary of Shanon-Nyquist sampling theorem that any sample of signal $\cos(2 \pi (f_0 + f_s) \, t)$ by sample rate $2 \pi f_s$ will result into the same series. Try it yourself by setting $t = k \, T_s$ and $T_s = \frac{2 \pi}{f_s}$.