= Solution
Put $s=\sin\theta$. By part (i), $\sin\phi=\sqrt2s$. The acute-angle geometry requires $0<s<1/\sqrt2$. Expressing the <tangents> in terms of their <sines> in the geometric constraint gives the required <implicit equation>
$$
\boxed{\frac{\sqrt3s}{\sqrt{1-s^2}}+\frac{\sqrt2s}{\sqrt{1-2s^2}}=2.}
$$
The hint suggests $s=1/m$ for an integer $m\geq2$. Testing $m=2$ gives
$$
\frac{\sqrt3/2}{\sqrt3/2}+\frac{1/\sqrt2}{1/\sqrt2}=1+1=2.
$$
Thus $\boxed{\sin\theta=1/2}$. Each term on the left is strictly increasing in $s$ over the allowed <interval>, so this solution is unique.
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