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 $y=\csc {x}$.

Resources

• Graphs of the Tangent, Cotangent, Cosecant and Secant Functions. Written notes created by Professor Esparza, UTSA.