Trigonometry / Unit 4: Graphs of Trigonometric Functions
4.6 Reciprocal and Cotangent Graphs
Notes
The graphs of $\csc x$, $\sec x$ and $\cot x$ come from their partners.
- $\displaystyle y = \csc x = \frac{1}{\sin x}$ has vertical asymptotes wherever $\sin x = 0$, that is at $x = n\pi$. Its range is $(-\infty, -1] \cup [1, \infty)$. Period $2\pi$.
- $\displaystyle y = \sec x = \frac{1}{\cos x}$ has asymptotes where $\cos x = 0$, at $\displaystyle x = \frac{\pi}{2} + n\pi$. Same range and period $2\pi$.
- $\displaystyle y = \cot x = \frac{\cos x}{\sin x}$ has period $\pi$ and asymptotes at $x = n\pi$.
- The "U" shapes touch the sine or cosine peaks and valleys.
Practice
Which line is a vertical asymptote of $y = \csc x$?
$\sin\pi = 0$, so $\csc\pi$ is undefined. At $\displaystyle \frac{\pi}{2}$ and $\displaystyle \frac{3\pi}{2}$ the graph touches $\pm 1$.
What is the range of $y = \sec x$?
Since $|\cos x| \le 1$, the reciprocal has $|\sec x| \ge 1$.
Which line is a vertical asymptote of $y = \cot x$?
$\displaystyle \cot x = \frac{\cos x}{\sin x}$ is undefined where $\sin x = 0$. At $\displaystyle \frac{\pi}{2}$ the value is 0.
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