Question

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

Short answer

The domain of the function is$R-\left\{n\mathrm{\pi }-\frac{\mathrm{\pi }}{4}\right\}$

Step-by-step solution

Step 1. Given Information

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

Step 2. Explanation

First simplify the given function,

$$\begin{gathered} y=\frac{3}{\sin x+\cos x} \\ y=\frac{3}{\sqrt{2}\left({\displaystyle \frac{1}{\sqrt{2}}}\sin x+{\displaystyle \frac{1}{\sqrt{2}}}\cos x\right)} \\ y=\frac{3}{\sqrt{2}\sin \left(x+{\displaystyle \frac{\mathrm{\pi }}{4}}\right)} \end{gathered}$$

The obtained function is not defined when the value sine function drops to zero. The function is not defined for $x=n\mathrm{\pi }-\frac{\mathrm{\pi }}{4}$, where n is any integer.

Thus, the domain of the function is $R-\left\{n\mathrm{\pi }-\frac{\mathrm{\pi }}{4}\right\}$where R is the set of all real numbers.