Graphs of the Tangent, Cotangent, Cosecant and Secant Functions

Tangent

Graph of tangent

The tangent of an angle is equivalent to the sine of that angle divided by the cosine of that angle.

$\tan x={\frac {\sin {x}}{\cos {x}}}$

The range of the function $y=\tan {x}$ is $(-\infty ,\infty )$. Note that since $\tan {x}={\frac {\sin {x}}{\cos {x}}}$, $y=\tan {x}$ 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 $y=\tan {x}$.

Cotangent

Graph of cotangent

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

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

The range of the function $y=\cot {x}$ 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 $y=\cot {x}$ 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 $y=\cot {x}$.

Secant

Graph of secant

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

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

Like $y=\cos {x}$, $y=\sec {x}$ is an even function; that is, $\sec {(x)}=\sec {(-x)}.Therangeofthefunction<math>y=\cos {x}$ 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, $y=\sec {x}$ 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.

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

Like $y=\sin {x}$, $y=\csc {x}$ is an odd function; that is, $\csc {(x)}=-\csc {(-x)}$. The range of the function $y=\csc {x}$ is the same as the range for $\sec {x}$ ($(-\infty ,-1]\cup [1,\infty )$). $y=\csc {x}$ 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 $y=\csc {x}$.

Resources

• Graphs of the Tangent, Cotangent, Cosecant and Secant Functions. Written notes created by Professor Esparza, UTSA.
• Trigonometry, WikiBooks: Pure Mathematics 2