Implicit Function Theorem Applied

Question

I am concerened I may have oversimplified my solution to this question.

My solution:

Let $F(x,y,z)=x-e^y\sin(z)$

By the implicit function theorem: $\displaystyle\frac{\partial z}{\partial x}=-\displaystyle\frac{\frac{\partial F}{\partial x}}{\frac{\partial F}{\partial z}}$

$$\frac{\partial F}{\partial z}=-e^y\cos(z)$$

As $e^y\neq0$ by the implicit function theorem $\cos(z)\neq0$

Answer 1

The implicit function theorem says that you can define $z$ as a function of $x$ and $y$ if and only if $\partial f/\partial z=-e^y\cos z\neq 0$, that is, $z/\pi\notin \Bbb Z$.