Graphs of the Tangent, Cotangent, Cosecant and Secant Functions

Tangent

File:Graph of tangent.png

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

File:Graph of cotangent.png

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

File:Graph of secant.png

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, $\sec {(x)}=\sec {(-x)}$. The range of the function $y=\cos {x}$ 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

File:Graph of cosecant.png

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

Licensing

Content obtained and/or adapted from:

• Trigonometry, WikiBooks: Pure Mathematics 2 under a CC BY-SA license