Question

The maximum valuue of the expression $\frac{1}{{\sin }^2\theta +3\sin \theta \cos \theta +5{\cos }^2\theta }$

Answer 1

Consider the problem

$\frac{1}{{\sin }^2\theta +3\sin \theta \cos \theta +5{\cos }^2\theta }$

$\begin{array}{c}=\frac{1}{\frac{1-\cos 2\theta }{2}+\frac{3}{2}\sin 2\theta +5\frac{1+\cos 2\theta }{2}}\\ =\frac{2}{6+3\sin 2\theta +4\cos 2\theta }\\ =\frac{2}{6+5\left(\frac{3}{5}\sin 2\theta +\frac{4}{5}\cos 2\theta \right)}\\ =\frac{2}{6+5\sin \left(2\theta +\alpha \right)}\end{array}$

Where, $\tan \alpha =\frac{4}{3}$, maximum when $\sin evalueis-1$

$\begin{array}{c}Value=\frac{2}{6-5}\\ =2\end{array}$