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$\newcommand{\ket}[1]{\left|{#1}\right\rangle}$ The $\tfrac{\theta}{2}$ can be seen on the picture given at the end of my answer. Maybe you are puzzled because you are tempting to interpret $$ \Psi=\cos\frac{\theta}{2} \lvert 0\rangle + e^{i\phi}\sin\frac{\theta}{2} \lvert 1\rangle $$ as a linear combination of vectors in the 3D Euclidean space. Note ...

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