Periodicity of sin 6πt, cos 8t and their sum

A CT sinusoid with angular frequency $\omega$ is always periodic with $T = 2\pi/\omega$. A sum is periodic only if $T_1/T_2$ is rational.

(i) $x(t) = \sin 6\pi t$, $\omega_1 = 6\pi$:

$$T_1 = \frac{2\pi}{6\pi} = \frac{1}{3}\ \text{s} \quad\Rightarrow\quad \text{periodic}$$

(ii) $y(t) = \cos 8t$, $\omega_2 = 8$:

$$T_2 = \frac{2\pi}{8} = \frac{\pi}{4}\ \text{s} \approx 0.785\ \text{s} \quad\Rightarrow\quad \text{periodic}$$

(iii) $x(t) + y(t)$:

$$\frac{T_1}{T_2} = \frac{1/3}{\pi/4} = \frac{4}{3\pi} \quad\text{(irrational)}$$

No integers $m, k$ give $mT_1 = kT_2$, so there is no common period.

Answer: $x(t)$ periodic, $T_1 = 1/3$ s; $y(t)$ periodic, $T_2 = \pi/4$ s; $x(t) + y(t)$ is not periodic (aperiodic).