Question

If \[\cot\theta = \frac{40}{9}\], find the values of cosecθ and sinθ.

Answer 1

\[\cot\theta = \frac{40}{9}\]           ...[Given]

We have,

\[{cosec}^2 \theta = 1 + \cot^2 \theta\]

\[ \Rightarrow {cosec}^2 \theta = 1 + \left( \frac{40}{9} \right)^2 \]

\[ \Rightarrow {cosec}^2 \theta = 1 + \frac{1600}{81} = \frac{1681}{81}\]

\[ \Rightarrow cosec \theta = \sqrt{\frac{1681}{81}} = \frac{41}{9}\]           ...[Taking square root of both sides]

Now,

\[\sin\theta = \frac{1}{cosec\theta}\]\[ \Rightarrow \sin\theta = \frac{1}{\frac{41}{9}}\]

\[ \Rightarrow \sin\theta = \frac{9}{41}\]

Thus, the values of cosecθ and sinθ are \[\frac{41}{9}\] and \[\frac{9}{41}\], respectively.