Chapter 7.2, Problem 5AYU

$y={\sec }^{-1}x$ means ________, where $\left|x\right|$ ______ and ______ $\le y\le$ ______, $y\ne \frac{\pi }{2}$.

To determine

To find: To convert the inverse secant, to find the inverse secant of a real number $x$ such that $\left|x\right|\ge 1$.

Answer to Problem 5AYU

Solution:

The inverse secant is converted into inverse cosine.

Explanation of Solution

Given:

Inverse secant of a real number $x$ such that $\left|x\right|\ge 1$.

Formula used:

$\text{Sec}x=1/\text{cos}x$

Calculation:

Domain of $\text{cos}x=$all real numbers. But $\text{cos}\left(\left(2n+1\right)\pi /2\right)=0$. $n=0,1,2\dots$

Range of $\text{cos}x=$All real numbers greater than or equal to 1 or less than or equal to $-1$.

If $y=\text{ inv cos}x$:

Then $x=\text{ cos}y$; $0\le y\le \pi$, $-1\le x\le 1$.

If $y=\text{ inv sec}x$:

Then $x=\text{ sec}y$; $0\le y\le \pi$, $y\ne \pi /2$; and $\left|x\right|\ge 1$.