Question : If $\sin A=\frac{2}{5}$, where $A$ is an acute angle, what is the value of $\frac{5 \sin A+2 \operatorname{cosec} A}{\sqrt{21} \sec A}$?
Option 1: $\frac{5}{4}$
Option 2: $\frac{7}{5}$
Option 3: $\frac{4}{5}$
Option 4: $\frac{5}{7}$
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Correct Answer: $\frac{7}{5}$
Solution :
$\sin A=\frac{2}{5}$
$⇒\cos A = \sqrt{1-\sin^2 A}$
$⇒\cos A = \sqrt{1-(\frac{2}{5})^2}$
$⇒\cos A = \sqrt{1-(\frac{4}{25})}$
$⇒\cos A = \sqrt{(\frac{21}{25})}$
$⇒\cos A = \frac{\sqrt{21}}{5}$
So, $\sec A = \frac{5}{\sqrt{21}}$
And, $\operatorname{cosec} A = \frac{5}{2}$
$\frac{5 \sin A+2 \operatorname{cosec} A}{\sqrt{21} \sec A}$
$=\frac{5 \times \frac{2}{5} +2 \times \frac{5}{2} }{\sqrt{21} \times \frac{5}{\sqrt{21}}}$
$=\frac{2 +5}{5}$
$=\frac{7}{5}$
Hence, the correct answer is $\frac{7}{5}$.
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