Question

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

Answer 1

Answer: 2

$\frac{1}{4{\cos }^2\theta +1+\frac{3}{2}\sin 2\theta }$
$\Rightarrow \frac{1}{2[1+\cos 2\theta ]+1+\frac{3}{2}\sin 2\theta }$
lies between $\frac{1}{2}$ to $\frac{11}{2}$
$\therefore$ maximum value is $2$
Minimum value of $1+4{\cos }^2\theta +3\sin \theta \cos \theta$
$1+\frac{4(1+\cos 2\theta )}{2}+\frac{3}{2}\sin 2\theta$
$=1+2+2\cos 2\theta +\frac{3}{2}\sin 2\theta$
$=3+2\cos 2\theta +\frac{3}{2}\sin 2\theta$
$\therefore =3-\sqrt{4+\frac{9}{4}}=3-\frac{5}{2}=\frac{1}{2}$
So maximum value of $\frac{1}{4{\cos }^2\theta +1+\frac{3}{2}\sin 2\theta }$ is $2$ .