= Solution
The $+y$ <eigenstate> is $|+y\rangle=(|\uparrow\rangle+i|\downarrow\rangle)/\sqrt2$, with $\sigma_y(1,i)^T=(1,i)^T$. Use the same duration $\tau_{\pi/2}=\pi/(2\Omega)$ but <phase>
$$
\varphi+\omega_0\tau_{\pi/2}=0\pmod{2\pi}.
$$
The relative amplitude is now $i$, giving
$$
\boxed{|\psi(\tau_{\pi/2})\rangle\sim|+y\rangle,\qquad
\langle s_y\rangle=\hbar/2.}
$$
The two transverse preparations differ by a quarter-cycle of RF <phase>, not by changing the <pulse area>. Reversing the desired transverse direction adds $\pi$ to the chosen <phase>.
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