Graphs of the Tangent, Cotangent, Cosecant and Secant Functions

Secant, Cosecant, and Cotangent

Secant

Graph of secant

The secant of an angle is the reciprocal of its cosine.

$\sec x={\frac {1}{\cos x}}$

The range of $\cos {x}$ is $[-1,1]$ , so the range of $\sec {x}$ is $(-\infty ,-1]\cup [1,\infty )$ , which is all possible outputs of 1/x for $-1\leq x\leq 1$ . Since ${\frac {1}{0}}$ is not defined, $\sec {x}$ is not defined whenever $\cos {x}=0$ ; that is, when $x=\pi k+{\frac {\pi }{2}}$ for any integer k. We can see that there is a vertical asymptote at each of these values in the graph of $y=\sec {x}$ .

Cosecant

Graph of cosecant

The cosecant of an angle is the reciprocal of its sine.

$\csc {x}={\frac {1}{\sin {x}}}$

The range of $\csc {x}$ is the same as the range for $\sec {x}$ ( $(-\infty ,-1]\cup [1,\infty )$ ). $\csc {x}$ is not defined whenever $\sin {x}=0$ ; that is, when $x=\pi k$ for any integer k. We can see that there is a vertical asymptote at each of these values in the graph of $y=\csc {x}$ .

Cotangent

Graph of cotangent

The cotangent of an angle is the reciprocal of its tangent.

$\cot x={\frac {1}{\tan x}}$

The range of $\cot {x}$ is $(-\infty ,\infty )$ , the same as the range of its reciprocal $\tan {x}$ . $\cot {x}={\frac {1}{\tan {x}}}={\frac {\cos {x}}{\sin {x}}}$ , so $\cot {x}$ is not defined whenever $\sin {x}=0$ ; that is, when $x=\pi k$ for any integer k. We can see that there is a vertical asymptote at each of these values in the graph of $y=\csc {x}$ .