Chapter 7.2, Problem 86AYU

To determine

To define: The function $y={\sec }^{-1}x$ and $y={\csc }^{-1}x$ .

Answer to Problem 86AYU

The parameter is a number used when equating the two vectors or equating coordinates of a vector lying on a line.

Explanation of Solution

Given information:

  

Definiton 1.

The inverse secant function is defined by restricting domain on secant function. The domain of secant function is

  $y=\sec x,[0,\frac{\pi }{2})\cup (\frac{\pi }{2},\pi ]$ whereas $y={\sec }^{-1}x,(-\infty ,-1]\cup [1,\infty )$ .

The inverse cosecant is the multivalued function with domain

  $y={\csc }^{-1}x,(-\infty ,-1]\cup [1,\infty )$ .

Definition 2.

  $y={\sec }^{-1}x$ means $x=\sec y$ where $\left|x\right|\ge 1$ and $0\le y\le \pi ,y\ne {\frac{\pi }{2}}^{*}$

  $y={\csc }^{-1}x$ means $x=\csc y$ where $\left|x\right|\ge 1$ and $-\frac{\pi }{2}\le y\le \frac{\pi }{2},y\ne 0^{†}$