Chapter 7.1, Problem 11AYU

True or False $\text{sin}\left({\text{sin}}^{-1}0\right)=0$ and $\text{cos}\left({\text{cos}}^{-1}0\right)=0$.

To determine

$\text{sin }\left({\text{sin}}^{-1}\text{0}\right)=\text{ 0; cos }\left({\text{cos}}^{\text{-1}}\text{0}\right)=\text{ 0}$. Check whether the given statement is true or false.

Answer to Problem 11AYU

Solution:

$\text{sin }\left({\text{sin}}^{-1}0\right)=0$ is True.

$\text{cos }\left({\text{ cos}}^{-1}0\right)=0$ is False.

Explanation of Solution

Given:

$\text{sin}\left({\text{sin}}^{-1}0\right)=0;\text{ cos}\left({\text{cos}}^{-1}0\right)=0$

Calculation:

$\text{sin}\left({\text{sin}}^{\text{-1}}\text{0}\right)\text{ = 0}$

$\text{Let }\theta ={\text{ sin}}^{\text{-1}}0$

$\text{Sin }\theta =0$

Value of $\theta$ for which sin $\theta =0$ is 0.

Therefore sin $\text{sin}\left({\text{sin}}^{-1}0\right)=0$ is true.

$\text{cos}\left({\text{cos}}^{-1}0\right)=0$

Let $\theta ={\text{ cos}}^{-1}0$

$\text{cos}\theta =0$

Value of $\theta$ for which $\text{cos}\theta =0$ is $\pi /2$.

Therefore $\text{cos}\left({\text{cos}}^{-1}0\right)=0$ is false.