"Sinusoidal" in DSP terminology
in my opinion, either of these two functions of time can be rightly labeled "sinusoidal":

$$ x(t) = A \sin(\omega t + \theta) $$

$$ y(t) = A e^{j(\omega t + \theta)} $$

where $A$, $\omega$ can be any non-zero real number (i wouldn't call DC a "sinusoid") and $\theta$ can be any real number.