Question

Find the value of the following trigonometric ratio :

\[ \cosec\left(-\frac{20\pi}{3}\right) \]

Solution

\[ \cosec\left(-\frac{20\pi}{3}\right) = \frac{1}{\sin\left(-\frac{20\pi}{3}\right)} \]

\[ = \frac{1}{\sin\left(-\frac{20\pi}{3}+6\pi\right)} \]

\[ = \frac{1}{\sin\left(-\frac{2\pi}{3}\right)} \]

Using,

\[ \sin(-\theta)=-\sin\theta \]

\[ = \frac{1}{-\sin\frac{2\pi}{3}} \]

\[ = \frac{1}{-\frac{\sqrt3}{2}} = -\frac{2}{\sqrt3} = -\frac{2\sqrt3}{3} \]

Answer :

\[ \boxed{ \cosec\left(-\frac{20\pi}{3}\right) = -\frac{2\sqrt3}{3} } \]