Question

Find the values of the following trigonometric ratio:

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

 

Answer 1

 We have: 

\[cosec\left( - \frac{20\pi}{3} \right) = cosec \left( - 1200^\circ \right)\]

\[cosec \left( - 1200^\circ \right) = - cosec \left( 1200^\circ \right) = - cosec \left( 90^\circ \times 13 + 30^\circ \right)\]

\[1200^\circ\text{ lies in the second quadrant in which the cosec function is positive . }\]

Also, 13 is an odd integer . 

\[ \therefore cosec\left( - 1200^\circ \right) = - cosec\left( 1200^\circ \right) = - cosec\left( 90^\circ \times 13 + 30^\circ \right) = - \sec30^\circ = - \frac{2}{\sqrt{3}}\]