Question

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

• A. $2$
• B. $3$
• C. $\frac{1}{2}$
• D. $\frac{1}{3}$

Answer 1

The correct option is A $2$

Ans. Ans. $\left(a\right)$.
$f\left(\theta \right)=\frac{1}{{\sin }^2\theta +3\sin \theta \cos \theta +5{\cos }^2\theta }$
$=\frac{1}{\frac{1-\cos 2\theta }{2}+\frac{3}{2}\sin 2\theta +\frac{5(1+\cos 2\theta )}{2}}$
$=\frac{2}{6+3\sin 2\theta +4\cos 2\theta }$
Hence, ${\left[f\left(\theta \right)\right]}_{maximum}=\frac{2}{6-5}=2$