The reciprocal trigonometric functions are defined from the three basic functions:

• $y=\sec x = \dfrac{1}{\cos x}$
• $y=\csc x = \dfrac{1}{\sin x}$
• $y=\cot x = \dfrac{1}{\tan x}$

Their graphs follow directly from the graphs of $\cos x$, $\sin x$ and $\tan x$. Wherever the original function equals zero, the reciprocal is undefined, producing a vertical asymptote. Wherever the original function equals $1$ or $-1$, the reciprocal also equals $1$ or $-1$, so the curves touch those points.

Graph of $y=\sec x$

The graph of $y=\sec x$ is symmetric in the $y$-axis and has period $360^\circ$, or $2\pi$ radians. It has vertical asymptotes at every value of $x$ for which $\cos x = 0$.

• Domain: $x \in \mathbb{R},\; x \neq 90^\circ, 270^\circ, 450^\circ,\dots$ or any odd multiple of $90^\circ$
• In radians: $x \in \mathbb{R},\; x \neq \dfrac{\pi}{2}, \dfrac{3\pi}{2}, \dfrac{5\pi}{2},\dots$ or any odd multiple of $\dfrac{\pi}{2}$
• This can also be written as $x \in \mathbb{R},\; x \neq \dfrac{(2n+1)\pi}{2},\; n \in \mathbb{Z}$
• Range: $y \leq -1$ or $y \geq 1$

Graph of $y=\csc x$

The graph of $y=\csc x$ has period $360^\circ$, or $2\pi$ radians. It has vertical asymptotes at all the values of $x$ for which $\sin x = 0$.

• Domain: $x \in \mathbb{R},\; x \neq 0^\circ, 180^\circ, 360^\circ,\dots$ or any multiple of $180^\circ$
• In radians: $x \in \mathbb{R},\; x \neq 0, \pi, 2\pi,\dots$ or any multiple of $\pi$
• This can also be written as $x \in \mathbb{R},\; x \neq n\pi,\; n \in \mathbb{Z}$
• Range: $y \leq -1$ or $y \geq 1$

Graph of $y=\cot x$

The graph of $y=\cot x$ has period $180^\circ$, or $\pi$ radians. It has vertical asymptotes at all the values of $x$ for which $\tan x = 0$.

• Domain: $x \in \mathbb{R},\; x \neq 0^\circ, 180^\circ, 360^\circ,\dots$ or any multiple of $180^\circ$
• In radians: $x \in \mathbb{R},\; x \neq 0, \pi, 2\pi,\dots$ or any multiple of $\pi$
• This can also be written as $x \in \mathbb{R},\; x \neq n\pi,\; n \in \mathbb{Z}$
• Range: $y \in \mathbb{R}$

Key features to remember

• Asymptotes occur where the denominator of the reciprocal is zero.
• Period matches that of the original trigonometric function: $2\pi$ for $\sec x$ and $\csc x$, and $\pi$ for $\cot x$.
• Symmetry: $y=\sec x$ is symmetric about the $y$-axis because $\cos x$ is even; $y=\csc x$ and $y=\cot x$ are not.
• Range: $\sec x$ and $\csc x$ never take values between $-1$ and $1$, whereas $\cot x$ can take any real value.