Picard's Big Theorem says that if a function $f(z)$ has an isolated singularity at a point $w$, then in every neighborhood of $w$, $f(z)$ hits every complex number infinitely many times, with perhaps at most one exception.

Can Picard's theorem be strengthened in the following fashion?

Let $V$ be an open disc such that $f(z)$ is holomorphic on $V - \lbrace w \rbrace$. Let $0 \leq \theta < \phi < 2\pi$, and define $Cone(w,V,\theta,\phi)$ to be $V \cap \lbrace w + re^{i\varphi} \mid r > 0, \theta < \varphi < \phi \rbrace$. Think of this as a "pizza slice" of the disc $V$.

Does $f(z)$ similarly hit every complex number (save perhaps one) infinitely many times in $Cone(w,V,\theta,\phi)$?

I am certain this is true, though I cannot prove it.