Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2002 / Paper 1 / 11 / c / Solution

Factor $e^{i\varphi /2}$ from the numerator and denominator:For $0<\varphi <\pi$, the tangent ranges over all positive real values. Thus $\boxed{\mathrm{Re}w=0,\mathrm{Im}w>0}$. Using the principal complex logarithm,The real part ranges over the entire real line, while the imaginary part is fixed: the locus is the horizontal line $\mathrm{Im}\log w=\pi /2$. Any other fixed logarithm branch gives the parallel line $\mathrm{Im}\log w=\pi /2+2\pi k$ for one integer $k$; allowing every branch gives the union of those lines.