Is there a function $ f $ such that $\sin(x)\cos(f(x))=a\cos(x)$?

Question

I'm working on some problem on differential geometry, where using the parametrization of the sphere given by $g(u,v)=(\sin(v)\cos(u),\sin(v)\sin(u),\cos(v))$ I want to find a curve $\lambda(t)$ on the domain of $g$ (let do not worry about what is that domain) such that $g(\lambda(t))$ is exactly the curve corresponding to the intersection of the plane $x=z\dfrac{\cos(\alpha)}{\sin(\alpha)}$ (where $\alpha\in(0,\pi)$ is fixed) with the sphere $\mathbb{S}^2\subset\mathbb{R}^{3}$. I already finded a parametrization that covers such a intersection, but I was not able to correspond this parametrization with $g$. So I began to try to find the solutions of the type (using $g$), $\sin(v)\cos(u)=a\cos(v)$ where $a=\dfrac{\cos(\alpha)}{\sin(\alpha)}$. Then came the idea that maybe can exists some function $u=f(v)$ such that $\sin(v)\cos(f(v))=a\cos(v)$ and if I find explicit such a $f$ then the curve I want in the domain of $g$ will be $(f(v),v)$. Can anyone help me?