$SU(2)$ consists of unitary $2\times2$ matrices of determinant one. Every element is unitarily diagonalizable with eigenvalues $e^{±iθ}$, and two are conjugate exactly when they have the same trace. Thus $t(g)=\operatorname{tr}(g)/2=\cosθ$ bijects conjugacy classes with $[-1,1]$.


Solution

Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 4 / 19I / b / i / Solution

 Solution (source code)