Past exam of the mathematical methods I course of the University of Oxford / 2017 resit / 4 / iii / Solution

The hyperbolic functions satisfy ${\cosh }^2\theta -{\sinh }^2\theta =1$, so a useful family isFor real $\theta$ this gives a one-real-parameter family of complex orthogonal matrices, which is enough to exhibit the requested hyperbolic parametrization. It does not parametrize every matrix in part (ii). For example $a=b=1/\sqrt{2}$ obeys the constraint but cannot have $a=\cosh \theta$ with real $\theta$, since then $a\ge 1$.

Allowing complex $\theta$ gives the full parametrization: $(a-ib)(a+ib)=1$, so choose a complex logarithm $\theta$ with $e^{\theta }=a-ib$. Then $a=\cosh \theta$ and $b=i\sinh \theta$. A general complex pair therefore needs a complex parameter; the real-parameter restriction in the question should not be mistaken for a completeness claim.