They're both periodic!

The function is $f(x) = \sin x + \sin \frac{\pi x}{180}$ . Since both its terms are periodic, the sum is periodic, no?

Hah, if only.

Assume the function is periodic with period $2p$ ; that is, $f(x-p) = f(x+p)$ for all real $x$ . Then

$\begin{aligned} \sin (x-p) + \sin \frac{\pi (x-p)}{180} &= \sin (x+p) + \sin \frac{\pi(x+p)}{180} \\ \sin (x+p) - \sin (x-p) &= \sin \frac{\pi(x+p)}{180} - \sin \frac{\pi (x-p)}{180} \\ 2 \cos x \sin p &= 2 \cos \frac{\pi x}{180} \sin \frac{\pi p}{180} \\ \end{aligned}$

This must be true for all real $x$ . In particular, this is true for $x = 0$ and $x = \frac{\pi}{2}$ . When $x = 0$ , the equation simplifies to $\sin p = \sin \frac{\pi p}{180}$ , and when $x = \frac{\pi}{2}$ , the left hand side becomes 0 but $\cos \frac{\pi x}{180} \neq 0$ , so the equation simplifies to $0 = \sin \frac{\pi p}{180}$ . Thus $\sin p = \sin \frac{\pi p}{180} = 0$ .

We know the roots of $\sin x = 0$ are $x = 2k\pi$ for any integer $k$ , so we have $p = 2k_1 \pi$ and $\frac{\pi p}{180} = 2k_2 \pi$ for some integers $k_1, k_2$ . Thus $p = 2\pi \cdot k_1 = 360 \cdot k_2$ . Since $p \neq 0$ (period can't be zero), we have $k_1, k_2 \neq 0$ , so we can compute $\frac{k_2}{k_1}$ ; this is $\frac{2\pi}{360} = \frac{\pi}{180}$ . But it is also a ratio of two integers, so it should be rational, contradiction (because $\pi$ is irrational while 180 is rational, the ratio can't be rational).

This contradiction arises because we assumed $p$ exists. Since it doesn't, the function therefore has no period and thus is not periodic.

$\sin{x}$ is periodic with $2\pi$

$\sin{\left( \pi \dfrac{x}{180} \right) }$ is periodic with $2\pi \times \dfrac{180}{\pi} = 360$

$f(x)$ is periodic by $LCM(2\pi,360)$ . Since the LCM does not exist , the function is not periodic .

I believe such a composite signal is only periodic if the quotient of the two signal frequencies is a rational number. This condition will not necessarily be satisfied if the domain for each signal frequency is the set of real numbers.