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The maximum value of the expression $ \dfrac{1}{{{\sin }^{2}}\theta +3\sin \theta \cos \theta +5{{\cos }^{2}}\theta } $ is
A. 0
B. 1
C. 2
D. 3
E. 4

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Last updated date: 19th Jun 2024
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Answer
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Hint: We need to find the minimum value of $ {{\sin }^{2}}\theta +3\sin \theta \cos \theta +5{{\cos }^{2}}\theta $ to find the maximum value of the expression $ \dfrac{1}{{{\sin }^{2}}\theta +3\sin \theta \cos \theta +5{{\cos }^{2}}\theta } $ . First, simplify the expression using trigonometric formulas. Then we try to convert the equation into an equation of singular ratio using the formula of $ \sin \left( A+B \right)=\sin A\cos B+\cos A\sin B $ . Then we apply the condition of minimum value to get the answer.

Complete step by step answer:
First, we try to find the simplified form of the equation $ {{\sin }^{2}}\theta +3\sin \theta \cos \theta +5{{\cos }^{2}}\theta $ .
We have the identity theorem of $ {{\sin }^{2}}\theta +{{\cos }^{2}}\theta =1 $ .
We convert the equation as
$ \Rightarrow {{\sin }^{2}}\theta +3\sin \theta \cos \theta +5{{\cos }^{2}}\theta $
$=\left( {{\sin }^{2}}\theta +{{\cos }^{2}}\theta \right)+\dfrac{3}{2}\left( 2\sin \theta \cos \theta \right)+4{{\cos }^{2}}\theta $
$ =1+\dfrac{3}{2}\left( 2\sin \theta \cos \theta \right)+2\left( 2{{\cos }^{2}}\theta \right) $
Now we apply the theorems of $ \sin 2\theta =2\sin \theta \cos \theta $ and $ 1+\cos 2\theta =2{{\cos }^{2}}\theta $.
$ \Rightarrow 1+\dfrac{3}{2}\left( 2\sin \theta \cos \theta \right)+2\left( 2{{\cos }^{2}}\theta \right) $
$ =1+\dfrac{3}{2}\left( \sin 2\theta \right)+2\left( 1+\cos 2\theta \right) $
$ =3+\dfrac{3\sin 2\theta }{2}+2\cos 2\theta $
To find the maximum value of the expression $ \dfrac{1}{{{\sin }^{2}}\theta +3\sin \theta \cos \theta +5{{\cos }^{2}}\theta } $ , we need to find the minimum value of $ {{\sin }^{2}}\theta +3\sin \theta \cos \theta +5{{\cos }^{2}}\theta $ which is equal to $ 3+\dfrac{3\sin 2\theta }{2}+2\cos 2\theta $ .
Now we try to convert the equation into one particular ratio.
We will take $\dfrac{5}{2} $ common from sin and cos terms and get
$ 3+\dfrac{3\sin 2\theta }{2}+2\cos 2\theta =3+\dfrac{5}{2}\left( \dfrac{3}{5}\sin 2\theta +\dfrac{4}{5}\cos 2\theta \right) $ .
We assume $ \cos \alpha =\dfrac{3}{5} $ which gives $ \sin \alpha =\sqrt{1-{{\left( \dfrac{3}{5} \right)}^{2}}}=\sqrt{1-\dfrac{9}{25}}=\sqrt{\dfrac{16}{25}}=\dfrac{4}{5} $ .
The equation becomes $ 3+\dfrac{5}{2}\left( \dfrac{3}{5}\sin 2\theta +\dfrac{4}{5}\cos 2\theta \right)=3+\dfrac{5}{2}\left( \sin 2\theta \cos \alpha +\cos 2\theta \sin \alpha \right) $ .
Now we apply $ \sin \left( A+B \right)=\sin A\cos B+\cos A\sin B $ . We assume $ A=2\theta ;B=\alpha $ .
\[3+\dfrac{5}{2}\left( \sin 2\theta \cos \alpha +\cos 2\theta \sin \alpha \right)=3+\dfrac{5}{2}\sin \left( 2\theta +\alpha \right)\].
Now we know that for any value of $ x\in \mathbb{R} $, the minimum value of $ \sin x $ is -1.
This means the minimum value of \[\sin \left( 2\theta +\alpha \right)\] is \[-1\]. We will find the minimum value of \[3+\dfrac{5}{2}\sin \left( 2\theta +\alpha \right)\] by putting minimum value of \[\sin \left( 2\theta +\alpha \right)\] as \[-1\].
So,\[3+\dfrac{5}{2}\sin \left( 2\theta +\alpha \right)=\]\[3+\dfrac{5}{2}\times(-1)\]
\[=\dfrac{1}{2}\]
Therefore, the minimum value of $ 3+\dfrac{3\sin 2\theta }{2}+2\cos 2\theta $ which is equal to $ {{\sin }^{2}}\theta +3\sin \theta \cos \theta +5{{\cos }^{2}}\theta $ , is $ \dfrac{1}{2} $ .
The maximum value of the expression $ \dfrac{1}{{{\sin }^{2}}\theta +3\sin \theta \cos \theta +5{{\cos }^{2}}\theta } $ is $ \dfrac{1}{\dfrac{1}{2}}=2 $ .
The correct option is (C).

Note:
The general formula of finding minimum and maximum value of equation types like $ a\sin \alpha +b\cos \alpha $ is $ -\sqrt{{{a}^{2}}+{{b}^{2}}}\le \left( a\sin \alpha +b\cos \alpha \right)\le \sqrt{{{a}^{2}}+{{b}^{2}}} $ . For any values of $ x\in \mathbb{R} $ , minimum and maximum values of ratios like $ \sin x,\cos x $ is $ -1\le \sin x,\cos x\le 1 $ .