Chapter 7.2, Problem 85AYU

Explain in your own words how you would use your calculator to find the value of ${\cot }^{-1}10$.

To determine

To find: The approximate value of inverse trigonometric function ${\text{cot}}^{-1}\left(10\right)$ using calculator.

Answer to Problem 85AYU

Solution:

The approximate value of inverse trigonometric function ${\text{cot}}^{-1}\left(10\right)\approx {0.089}^c$.

Explanation of Solution

Given:

Inverse trigonometric function is given as ${\text{cot}}^{-1}\left(10\right)$.

Calculation:

Let $\theta ={\text{ cot}}^{-1}\left(10\right)$

$\text{cot}\theta =10$

The range of the inverse trigonometric function ${\text{cot}}^{-1}\left(10\right)$ lies within $0<\theta <\pi$.

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

Therefore to find ${\text{cot}}^{-1}\left(10\right)$, we cannot directly find the value in the calculator.

We need to find equivalent ${\text{cos}}^{-1}$ value.

Steps to find ${\text{cos}}^{-1}x$,

$\text{cot}\theta =10=x/y$

where $x=10$ and $y=1$.

Defining $\text{cos}\theta =x/r$.

Where $r=\text{ radius of the circle}$.

To calculate $r$ by equation,

$r^2 =x^2 +y^2$

$r=\text{ op }=\surd \left({10}^2+1^2\right)=\surd 101$

Since $x$ and $y$ are positive, $\theta$ lies in the I

Quadrant.

$\text{cos}\theta =x/r$

$\text{cos}\theta =10/\surd 101$

Using calculator, we need to find $10/\surd 101=0.996$.

Then we need to keep the calculator in the radian mode.

$\text{cos}\theta =0.996$

therefore, $\theta ={\text{ cos}}^{-1}\left(0.996\right)$.

$\theta ={\text{ cot}}^{-1}\left(10\right)={\text{ cos}}^{-1}\left(10/\surd 101\right)={\text{ cos}}^{-1}\left(0.996\right)$

$\theta \approx {0.089}^c$

The same $\theta$ can be obtained if the calculator is kept in the degree mode.

$\theta \approx 5.12°$