Cosecant And Secant Graphs | Brilliant Math & Science Wiki

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From the definition of the secant and cosecant functions, we have

\[\sec( \theta) = \frac{1}{\cos(\theta)},\quad \csc( \theta) = \frac{1}{\sin(\theta)}.\]

This shows \(\sec(\theta)\) is not defined for values of \(\theta\) such that \(\cos(\theta) = 0\). From the graph of the cosine function we see that \(\cos(\theta) = 0\) when \(\theta = \frac{\pi}{2} + k\pi\) for any integer \(k\), which implies that the secant function has vertical asymptotes at these values of \(\theta\). The graph of the secant function is as follows:

Similarly, \(\csc(\theta)\) is not defined for values of \(\theta\) such that \(\sin(\theta) = 0\), which occurs for \(\theta = 0+ k\pi\) for any integer \(k\). Therefore, the cosecant function has vertical asymptotes at these values of \(\theta\). The graph of the cosecant function is as follows:

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