# Chapter 6 - Inverse Circular Functions and Trigonometric Equations - Section 6.1 Inverse Circular Functions - 6.1 Exercises - Page 264: 35

$\displaystyle \csc^{-1}\frac{\sqrt{2}}{2}$ is not defined.

#### Work Step by Step

Inverse Cosecant Function: $y=\csc^{-1}x$ or $y=$ arccsc $x$ means that $x=\csc y$, for $-\displaystyle \frac{\pi}{2} \leq y \leq \frac{\pi}{2}$ , $y\neq 0$. ----------------- If there exists a y such that $\displaystyle \csc y=\frac{\sqrt{2}}{2}$, then, sine being reciprocal to csc, $\displaystyle \sin y=\frac{2}{\sqrt{2}}\cdot\frac{\sqrt{2}}{\sqrt{2}}=\frac{2\sqrt{2}}{2}=\sqrt{2}$, which can not be for any y, since the range of sine is $[-1,1].$ So, $\displaystyle \csc^{-1}\frac{\sqrt{2}}{2}$ is not defined.

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