Past exam of the mathematics course of the University of Cambridge / 2009 / ib / Paper 4 / 4E (Complex Analysis)

Rouché's theorem states that if $f,g$ are holomorphic on and inside a simple closed contour and $|g|<|f|$ on it, then $f$ and $f+g$ have the same number of zeros inside, counted with multiplicity. On $|z|=1$, the power series for sine givesApply Rouché's theorem with $f=2z$ and $g=-\sin z$. There is one zero inside and none on the boundary. Since $z=0$ is a zero and $(2z-\sin z{)}^{\prime }{|}_{z=0}=1$, it is simple. Exactly one complex number qualifies: $z=0$.