What is the unit of measurement of $z$ as a solution to the equation: $\sin z=2$?

Question

Recently, I uploaded a video on my channel solving the equation $\sin z=2$ See

Now, one question that I was asked is "What is the unit of this $Z$ as a solution to the equation: $\sin Z=2$. Is it in Radian? Can I change it to Degree?... And most importantly, what is the physical significance of the solution?" Though based on my limited understanding and knowledge, I answered the question, but I seek a concrete answer for the same.

I will include my answer for your perusal "In complex trigonometric functions, $\sin z$ is expressed as power series and not as a measure of angles on a circle in classical trigonometry. So, neither of the unit suffice in this case and which is also evident from the answer which is in the form of $x+Iy$. Another representation of the same could be in terms of hyperbolic sines and cosines where unit of measurement is hyperbolic than circular."

Answer 1

From $e^{iz} = \cos(z)+i\sin(z)$ we get $\sin(z) = (e^{iz} -e^{-iz})/2$.

Solve $(x-1/x)/2 =2$ and then $e^{iz} = x$.

All well known, of course.

Answer 2

The question is exactly the same as asking, "what is the unit of $x$ in $x^{2}=2$?"

The student might be tempted to argue something like "the unit is obviously meters, and the result is an area". But this is not so obvious at all: in fact, people often speak of "seconds-squared", $s^2,$ or "per second", $s^{-1};$ so we see that the $x$ in $x^{2}$ need not be a length, it could be a duration.

What the unit is in a particular mathematical model depends only on what $x$ is being used to represent in that model. If $x$ represents a distance, then the unit should be a unit of distance, like meters; but there is no fixed choice of unit, so the viewer's question is missing the context necessary to give it an answer.

In practice, complex numbers are often useful merely as computational aids, or as a framework to make nice mathematical models, and then the real and imaginary parts are associated with units of their own. One place where complex numbers themselves famously take centre-stage in a physical model is in quantum mechanics: the wave function outputs complex numbers, and the square magnitude of these complex numbers represents probability density. But this is a pretty esoteric example, to be honest; and even in this example, the "unit" of the complex number (if that's really something you want to discuss) depends on how many dimensions your model is meant to account for.