Browse Questions

# Find the principal values of the following: $cosec^{-1} (2)$

$\begin{array}{1 1} -\frac{\pi}{6} \\ \frac{\pi}{2} \\ -\frac{\pi}{2} \\ \frac{\pi}{6} \end{array}$

Toolbox:
• The range of the principal value of $\; cosec^{-1}x$ is $\left [ \large-\frac{\pi}{2}, \large\frac{\pi}{2} \right ] -${$0$}
Let $cosec^{-1}2 = x \Rightarrow cosec (x) = 2$
We know the range of the principal value of $\; cosec^{-1}x$ is $\left [ \large-\frac{\pi}{2}, \large\frac{\pi}{2} \right ] -${$0$}
Therefore, $cosec (x) = 2 = cosec \large\frac{\pi}{6}$
$\Rightarrow x=\large\frac{\pi}{6}$, where $x \;\epsilon\;$ $\left [ \large-\frac{\pi}{2}, \large\frac{\pi}{2} \right ] -${$0$}
Hence the principal value of $\; cosec^{-1}2$ is $\large\frac{\pi}{6}$

edited Mar 15, 2013