The maximum value of the expression 1 sin 2 + 3 sin cos + 5 cos 2 is _______. [2010]

Question

The maximum value of the expression $\frac{1}{\sin^{2} θ + 3 \sin θ \cos θ + 5 \cos^{2} θ}$ is _______. [2010]

Smart Achievers · Accepted Answer

(2)

Let $f (θ) = \frac{1}{g (θ)},$

where $g (θ) = \sin^{2} θ + 3 \sin θ \cos θ + 5 \cos^{2} θ$

$Clearly f is maximum when g is minimum$

$Now g (θ) = \frac{1 - \cos 2 θ}{2} + \frac{3}{2} \sin 2 θ + \frac{5}{2} (1 + \cos 2 θ)$

$= 3 + 2 \cos 2 θ + \frac{3}{2} \sin 2 θ$ $\geq 3 + (- \sqrt{4 + \frac{9}{4}})$

$∴$ $g_{\min} = 3 - \frac{5}{2} = \frac{1}{2} \Rightarrow f_{\max} = 2$

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