(a) For \(\cos\theta\ne0\), \[3\tan^2\theta+5\sin^2\theta=\frac{3\sin^2\theta}{1-\sin^2\theta}+5\sin^2\theta=\frac{8\sin^2\theta-5\sin^4\theta}{1-\sin^2\theta}.\]

(b) Put \(u=\sin^2\theta\). Then \[8u-5u^2=9(1-u)\Rightarrow5u^2-17u+9=0\Rightarrow u=\frac{17\pm\sqrt{109}}{10}.\] Reject the value greater than 1. With \(u=(17-\sqrt{109})/10\), the acute angle is \(54.1^\circ\). All permitted angles are \(54.1^\circ,125.9^\circ,234.1^\circ\).