Question : If $\cos \theta=\frac{9}{13}$, then what is the value of $\operatorname{cosec} \theta$?
Option 1: $\frac{13}{\sqrt{22}}$
Option 2: $\frac{13 \sqrt{22}}{44}$
Option 3: $\frac{2 \sqrt{22}}{13}$
Option 4: $\frac{\sqrt{22}}{13}$
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Correct Answer: $\frac{13 \sqrt{22}}{44}$
Solution :
Given,
$\cos\theta = \frac{9}{13}$
We know, $\sin^2\theta+\cos^2\theta=1$
⇒ $\sin θ = \sqrt{(1 - \cos^2θ)}$
⇒ $\sin θ = \sqrt{(1 - (\frac{9}{13})^2)}$
⇒ $\sin θ = \sqrt{1 - \frac{81}{169}}$
⇒ $\sin θ = \sqrt{\frac{88}{169}}=\frac{2\sqrt{22}}{13}$
⇒ $\operatorname{cosec} θ = \frac{1}{\sin θ}=\frac{13}{2\sqrt{22}} = \frac{13\sqrt{22}}{44}$
Hence, the correct answer is $\frac{13\sqrt{22}}{44}$.
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