Exercise 1.2.5 (5-6)

Question

Find the domain of the function.$$5.\quad f(x)=\frac{\cos x}{1-\sin x}\qquad \quad 6.\quad g(x)=\frac{1}{1-	an x}.$$

Khagan Gulusoy · Accepted Answer

\begin{enumerate}
    \item [5.] The denominator of a fraction cannot be equal to $0.$ Thus, we have:
    $$\forall x\in \mathbb{R}:\quad 1-\sin x
eq 0\iff \sin x
eq 1\iff x
eq \frac{\pi}{2}+2\pi n.$$
    Therefore, the domain of the function $f$ is:
    $$\left\{x\in \mathbb{R}\mid x
eq \frac{\pi}{2}+2\pi n	ext{ for some }n\in \mathbb{N}ight\}.$$
    \item [6.] The denominator of a fraction cannot be equal to $0.$ Thus, we have:
    $$\forall x\in \mathbb{R}:\quad 1-	an x
eq 0\iff 	an x
eq 1\iff x
eq \frac{\pi}{4}+\pi n.$$    
    Therefore, the domain of the function $g$ is:
    $$\left\{x\in \mathbb{R}\mid x
eq \frac{\pi}{4}+\pi n	ext{ for some }n\in \mathbb{N}ight\}.$$

\end{enumerate}

Exercise 1.2.5 (5-6)

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Exercise 1.2.5 (5-6)

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