Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2024 / Paper 1 / 11 / e / Solution

Since $x=\cos y$ with real $y$, necessarily $-1\le x\le 1$. Use Euler's formula and set $w=e^{iy}\ne 0$. Thenso the quadratic equation gives $w=x\pm i\sqrt{1-x^2}$. Let $q=x+i\sqrt{1-x^2}$. It lies on the upper half of the unit circle, so $q=e^{i\theta }$ with $0\le \theta \le \pi$. Choose the complex logarithm value $\ln q=i\theta$, taking $\ln (-1)=i\pi$ at $x=-1$. The two roots are $e^{\pm i\theta }$, so $e^{iy}=e^{\pm i\theta }$ gives $y=\pm \theta +2n\pi$.

Because $i\ln q=-\theta$, this set of real solutions is exactlyThe formula includes $x=\pm 1$: the two sign families then coincide after adjusting $n$. No real $y$ exists if $|x|>1$.