Which of the following is an asymptote of $y=\csc (x)$?
To identify the vertical asymptotes of the function \(y = \csc(x)\), we first recall that the cosecant function is defined as the reciprocal of the sine function:
\[y = \csc(x) = \frac{1}{\sin(x)}\]
A vertical asymptote occurs where the function is undefined, which, for this rational-like trigonometric expression, happens where the denominator is zero. Thus, we set \(\sin(x) = 0\). The values of \(x\) for which \(\sin(x) = 0\) occur at integer multiples of \(\pi\):
\[x = n\pi, \quad n \in \mathbb{Z}\]
This means the vertical asymptotes of \(y = \csc(x)\) are located at \(..., -2\pi, -\pi, 0, \pi, 2\pi, ...\).
We evaluate the given options to see which corresponds to the form \(x = n\pi\):
Therefore, the only provided value that corresponds to a vertical asymptote is \(x = -\pi\).
Based on our analysis, the vertical asymptotes of the cosecant function are located at integer multiples of \(\pi\). Comparing our options, \(x = -\pi\) is the only one that satisfies this condition. Thus, the correct asymptote from the list is \(x = -\pi\).
Vertical asymptotes represent lines where a function grows without bound. In electronics, when analyzing the impedance or resonance of circuits involving capacitors and inductors, trigonometric functions often appear in equations. Identifying these asymptotes is crucial for engineers to predict where the circuit might become unstable or where the signal voltage might approach infinity, allowing them to design safer and more efficient systems.