Find the principal value of ${c o s e c}^{- 1} (- \sqrt{2})$

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Solution

Let ${c o s e c}^{- 1} (- \sqrt{2}) = y$

$\Rightarrow c o s e c y = - \sqrt{2} = - c o s e c (\frac{π}{4}) = c o s e c (- \frac{π}{4})$ .

We know that the range of the principal value of ${c o s e c}^{- 1}$ is $[- \frac{π}{2}, \frac{π}{2}] - {0}$ and $c o s e c (- \frac{π}{4}) = - \sqrt{2}$

Therefore, the principal value of ${c o s e c}^{- 1} (- \sqrt{2})$ is $- \frac{π}{4} .$

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