Chapter 7.2, Problem 48AYU

In Problems 45-56, use a calculator to find the value of each expression rounded to two decimal places.

${\sec }^{-1}\left(-3\right)$

To determine

To find: The approximate value of inverse trigonometric function ${\text{sec}}^{-1}\left(-3\right)$ using calculator and rounding to 2 decimal places.

Answer to Problem 48AYU

The approximate value of inverse trigonometric function ${\text{sec}}^{-1}\left(-3\right)$ to 2 decimal places $\approx 1.90$.

Explanation of Solution

Given:

Inverse trigonometric function ${\text{sec}}^{-1}\left(-3\right)$.

Calculation:

Let $\theta = {\text{sec}}^{-1}\left(-3\right)$.

$\text{secθ }=-3$; $0\le \theta \le \pi$, $\theta \ne \pi /2$

$y = {\text{cos}}^{-1}x$ has the same range as $y = {\text{sec}}^{-1}x$ except where undefined.

Using the formula,

$\text{sec}\theta =1/\text{cos}\theta = -3$

$\text{cos}\theta =1/-3=-0.333$

$\Theta ={\text{ cos}}^{-1}\left(1/-3\right)={\text{ cos}}^{-1}\left(-1/3\right)$

$\Theta ={\text{ cos}}^{-1}\left(-0.33\right)$

Keeping the calculator in radian mode,

${\text{sec}}^{-1}\left(-0.33\right)= {\text{cos}}^{-1}\left(-0.33\right) \approx 1.90$