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Find the principal values of the following: \[ cosec^{-1} (-\sqrt 2) \]

$\begin{array}{1 1} \large {\frac{-\pi}{4}} \\ \large {\frac{-\pi}{2}} \\ \large {\frac{-\pi}{8}} \\ \large {\frac{-\pi}{6}} \end{array} $

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Ans : \( cosec^{-1}cosec \bigg( \large\frac{-\pi}{4} \bigg) = \large\frac{-\pi}{4} \)

 

answered Feb 22, 2013 by thanvigandhi_1
edited Mar 15, 2013 by thanvigandhi_1
 
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  • The range of the principal value of $\; cosec^{-1}x$ is $\left [ -\large\frac{\pi}{2}, \large\frac{\pi}{2} \right ] -${$0$}
Let $cosec^{-1}-\sqrt2 = x \Rightarrow cosec (x) =-\sqrt 2$
 
The range of the principal value of $\; cosec^{-1}x$ is $\left [ -\large\frac{\pi}{2}, \large\frac{\pi}{2} \right ] -${$0$}
\( \therefore\) $cosec (x) = -\sqrt 2 = cosec \large\frac{-\pi}{4}$
$\Rightarrow x=\large\frac{-\pi}{4}$, where $x \;\epsilon\;$ $\left [ -\large\frac{\pi}{2}, \large\frac{\pi}{2} \right ] -${$0$}
 
Hence the principal value of $\; cosec^{-1}(-\sqrt 2)$ is $\large\frac{-\pi}{4}$

 

answered Mar 2, 2013 by balaji.thirumalai
edited Mar 15, 2013 by thanvigandhi_1
 
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