Is sin(t) periodic; energy or power

Periodicity

$\sin(t+T) = \sin t$ for all $t$ when $T = 2\pi m$. The smallest positive value is

$$T_0 = \frac{2\pi}{\omega_0} = \frac{2\pi}{1} = 2\pi\ \text{s}$$

So $x(t) = \sin t$ is periodic with $T_0 = 2\pi$.

Energy or power

Energy: $E = \int_{-\infty}^{\infty}\sin^2 t\,dt = \infty$ (the area of each period adds up forever).

Power (average over one period):

$$\begin{aligned} P &= \frac{1}{2\pi}\int_0^{2\pi}\sin^2 t\,dt = \frac{1}{2\pi}\int_0^{2\pi}\frac{1-\cos 2t}{2}\,dt \\ &= \frac{1}{2\pi}\cdot\frac{2\pi}{2} = \frac{1}{2} \end{aligned}$$

Answer: $\sin t$ is periodic ($T_0 = 2\pi$ s) and is a power signal with $P = 0.5$ W (normalised), $E = \infty$.