Throughout this problem, t∈R is measured in seconds

Throughout this problem, \(t\in\mathbb{R}\) is measured in seconds. A continuous-time sinusoid is periodic with fundamental period \(T_0>0\) if \(T_0\) is the smallest positive number satisfying \(x(t+T_0)=x(t)\) for every \(t\); if no such \(T_0\) exists, the signal is not periodic. (a) Let \(x(t)=7\cos\bigl((12\pi/5)t+\pi/7\bigr)\). Determine whether \(x\) is periodic. If it is, determine its fundamental period in seconds and its fundamental frequency in hertz. (b) Let \(y(t)=5\cos(t)+\cos(\sqrt{2}\, t)\). Determine whether \(y\) is periodic. If it is not, prove that a common period would force \(\sqrt{2}\) to be rational. (c) Let \(z(t)=4\sin(6\pi t)-4\sin(6\pi t+\pi)\). Simplify \(z(t)\) using a trigonometric identity, then determine whether \(z\) is periodic and, if so, its fundamental period.

A single continuous-time sinusoid is always periodic; a sum of two is periodic only when a common period exists, i.e. when the period ratio is rational.