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

Question

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

Aicharya · Accepted Answer

Let $ \begin{aligned} & f(	heta)=\frac{1}{\sin ^2 	heta+3 \sin 	heta \cos 	heta+5 \cos ^2 	heta} \ & 	ext { Again let } \ & g(	heta)=\sin ^2 	heta+3 \sin 	heta \cos 	heta+5 \cos ^2 	heta \ & =\frac{1-\cos 2 	heta}{2}+5\left(\frac{1+\cos 2 	heta}{2}ight)+\frac{3}{2} \sin 2 	heta \ & =3+2 \cos 2 	heta+\frac{3}{2} \sin 2 	heta \ & 	herefore \quad g(	heta)_{\min }=3-\sqrt{4+\frac{9}{4}}=3-\frac{5}{2}=\frac{1}{2} \ & 	herefore \quad f(	heta)=\frac{1}{g(	heta)_{\min }}=2 \end{aligned} $

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

Solution