 Solution

The nonnegative modulus $r=|z|$ must be a zero of the sine function. The restriction $r<10$ permits $r=0,\pi ,2\pi ,3\pi$, since $3\pi <10<4\pi$. Thus the locus is the origin and three concentric circles, centred at the origin with radii $\pi$, $2\pi$ and $3\pi$. A circle of radius zero is just one point.