 Solution

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