Chapter 10, Problem 10CR

To determine

The domain of the function $f\left(x\right)=\frac{3}{\sin x+\cos x}$

Answer to Problem 10CR

Solution:

The domain of $f\left(x\right)=\frac{3}{\sin x+\cos x}$ is $\left\{x|x\ne \frac{3\pi }{4}+\pi n;n\in \mathbb{Z}\right\}$

Explanation of Solution

Given information:

The function $f\left(x\right)=\frac{3}{\sin x+\cos x}$

Consider the function $f\left(x\right)=\frac{3}{\sin x+\cos x}$

Domain is the set of input values at which the function is defined.

The given function is a rational function which is not defined when the denominator becomes zero.

That is, the function is not defined, when $\sin x+\cos x=0$

Consider $\sin x+\cos x=0$

Divide both sides by $\cos x$

  $\frac{\sin x}{\cos x}+\frac{\cos x}{\cos x}=\frac{0}{\cos x}$

  $\Rightarrow \tan x+1=0$

  $\Rightarrow \tan x=-1$

  $\Rightarrow x=\frac{3\pi }{4}+\pi n;n$ is an integer

Therefore, the domain of $f\left(x\right)=\frac{3}{\sin x+\cos x}$ is $\left\{x|x\ne \frac{3\pi }{4}+\pi n;n\in \mathbb{Z}\right\}$