= Solution
The required state is the maximal-<spin coherent state> pointing at azimuth $\varphi$ in the equatorial plane. In the same ordered $S_z$ basis it is
$$
\boxed{
|\mathbf n,3/2\rangle
=\frac1{2\sqrt2}
\begin{pmatrix}
1\\
\sqrt3e^{i\varphi}\\
\sqrt3e^{2i\varphi}\\
e^{3i\varphi}
\end{pmatrix}.}
$$
Its norm is $(1+3+3+1)/8=1$. Direct multiplication using the matrices in part (b), or rotation of $|\hat{\mathbf z},3/2\rangle$, gives
$$
(S_x\cos\varphi+S_y\sin\varphi)|\mathbf n,3/2\rangle
=\frac32\hbar|\mathbf n,3/2\rangle.
$$
It therefore yields the value $3\hbar/2$ with probability one when spin is measured along $\mathbf n$.
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