Question

How to find the value of $\mathrm{cosec}\left({210}^{°}\right)$?

Answer 1

Find the value of given trigonometric function

First, find the value of $\sin \left({210}^{°}\right)$

We know that

$\sin \left(\alpha +\beta \right)=\sin \left(\alpha \right)\cos \left(\beta \right)+\cos \left(\alpha \right)\sin \left(\beta \right)$

Now by substituting the value of $\alpha ={180}^{°}$ and $\beta ={30}^{°}$in the trigonometric identity we get

$$\begin{gathered} \sin \left({180}^{°}+{30}^{°}\right)=\sin \left({180}^{°}\right)\cos \left({30}^{°}\right)+\cos \left({180}^{°}\right)\sin \left({30}^{°}\right) \\ \Rightarrow \sin \left({180}^{°}+{30}^{°}\right)=0-1\times \frac{1}{2}\left[\because \cos \left({180}^{°}\right)=-1,\sin \left({180}^{°}\right)=0,\cos \left({30}^{°}\right)=\frac{\sqrt{3}}{2}\mathrm{and}\sin \left({30}^{°}\right)=\frac{1}{2}\right] \\ \Rightarrow \sin \left({210}^{°}\right)=-\frac{1}{2} \end{gathered}$$

Now, find the value of $\mathrm{cosec}\left({210}^{°}\right)$

$$\begin{gathered} \mathrm{cosec}\left({210}^{°}\right)=\frac{1}{\sin \left({210}^{°}\right)}\left[\because \mathrm{cosec}\left(x\right)=\frac{1}{\sin \left(x\right)}\right] \\ \mathrm{cosec}\left({210}^{°}\right)=-2 \end{gathered}$$

Hence the value of $\mathrm{cosec}\left({210}^{°}\right)$ is $-2$.