Question

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

Answer 1

We have,

$$\begin{gathered} {\mathrm{cosec}}^2\theta =1+{\cot }^2\theta \\ \Rightarrow {\mathrm{cosec}}^2\theta =1+{\left(\frac{40}{9}\right)}^2 \\ \Rightarrow {\mathrm{cosec}}^2\theta =1+\frac{1600}{81}=\frac{1681}{81} \\ \Rightarrow \mathrm{cosec}\theta =\sqrt{\frac{1681}{81}}=\frac{41}{9} \end{gathered}$$
Now,

$$\begin{gathered} \sin \theta =\frac{1}{\mathrm{cosec}\theta } \\ \Rightarrow \sin \theta =\frac{1}{{\displaystyle \frac{41}{9}}} \\ \Rightarrow \sin \theta =\frac{9}{{\displaystyle 41}} \end{gathered}$$
Thus, the values of cosecθ and sinθ are $\frac{41}{9}$ and $\frac{9}{41}$, respectively.