Question

graph $y = \frac{1}{6} \csc \theta$.

Explanation:

Step 1: Recall the relationship between \(y = A\csc\theta\) and \(y = A\sin\theta\)

The general form of the cosecant function is \(y = A\csc\theta=\frac{A}{\sin\theta}\). The vertical asymptotes of \(y = A\csc\theta\) occur where \(\sin\theta = 0\), i.e., \(\theta = n\pi\), \(n\in\mathbb{Z}\). The period of \(y=\csc\theta\) is \(2\pi\). For \(y = \frac{1}{6}\csc\theta\), the amplitude - like factor (related to the reciprocal of the sine function's amplitude) is \(|A|=\frac{1}{6}\).

Step 2: Analyze the graph based on key features

The function \(y=\frac{1}{6}\csc\theta\) has vertical asymptotes at \(\theta = 0,\pm\pi,\pm2\pi,\cdots\). The graph of \(y = \frac{1}{6}\csc\theta\) is composed of two - part curves (branches) between consecutive vertical asymptotes. The "U - shaped" parts (for positive \(y\)) and the "inverted U - shaped" parts (for negative \(y\)) are scaled by a factor of \(\frac{1}{6}\) compared to the basic \(y = \csc\theta\) graph (\(y=\csc\theta\) has values \(y\geq1\) or \(y\leq - 1\), while \(y=\frac{1}{6}\csc\theta\) has values \(y\geq\frac{1}{6}\) or \(y\leq-\frac{1}{6}\)).

Since the second graph (the one with more vertical asymptotes at \(\theta = n\pi\), \(n\in\mathbb{Z}\)) matches the key features of \(y=\frac{1}{6}\csc\theta\) (vertical asymptotes at \(\theta = n\pi\) and the correct scaling factor), it is the correct graph.

Answer:

The second graph (the one with vertical asymptotes at \(\theta=0,\pm\pi,\pm2\pi,\cdots\)) is the graph of \(y = \frac{1}{6}\csc\theta\).