Graphs of Tangent and Cotangent Functions

Graphs of Tangent & Cotangent

Graphing the Tangent Function

The tangent function, tan(x), is a fundamental trigonometric function with unique graphing properties. Its graph is periodic and features vertical asymptotes where the function is undefined.

• Definition: The tangent function is defined as $\tan(x) = \frac{\sin(x)}{\cos(x)}$.

• Periodicity: The function repeats every $\pi$ units, so $\tan(x + \pi) = \tan(x)$.

• Asymptotes: Vertical asymptotes occur where $\cos(x) = 0$, i.e., at $x = \frac{\pi}{2} + k\pi$, where $k$ is any integer.

• Key Points: The graph passes through the origin (0,0) and is symmetric about this point.

Table of Values for $\tan(x)$:

x	0	$\frac{\pi}{4}$	$\frac{\pi}{2}$	$\frac{3\pi}{4}$	$\pi$
$\tan(x)$	0	1	undefined	-1	0

Transformations: The general form $y = a \tan(bx - c) + d$ allows for vertical stretching/compression, period changes, phase shifts, and vertical shifts.

• Period: $\text{Period} = \frac{\pi}{|b|}$

• Phase Shift: $\text{Phase Shift} = \frac{c}{b}$

• Vertical Shift: $d$ shifts the graph up or down.

Example: Graph $y = \tan(2x)$

• Period: $\frac{\pi}{2}$

• Vertical asymptotes at $x = \frac{\pi}{4} + k\frac{\pi}{2}$

Practice Problem

Given the graph of $y = \tan(bx)$, determine the value of $b$ by analyzing the period and location of asymptotes.

• If the period is $\frac{\pi}{3}$, then $b = 3$.

Graphing the Cotangent Function

The cotangent function, cot(x), is the reciprocal of the tangent function and has its own distinct graphing characteristics.

• Definition: $\cot(x) = \frac{\cos(x)}{\sin(x)}$

• Periodicity: The function repeats every $\pi$ units, so $\cot(x + \pi) = \cot(x)$.

• Asymptotes: Vertical asymptotes occur where $\sin(x) = 0$, i.e., at $x = k\pi$, where $k$ is any integer.

• Key Points: The graph decreases from $+\infty$ to $-\infty$ between asymptotes.

Table: Comparison of Tangent and Cotangent

Function	Period	Asymptotes
$\tan(x)$	$\pi$	$x = \frac{\pi}{2} + k\pi$
$\cot(x)$	$\pi$	$x = k\pi$

Transformations: The general form $y = a \cot(bx - c) + d$ allows for similar transformations as the tangent function.

• Period: $\frac{\pi}{|b|}$

• Phase Shift: $\frac{c}{b}$

• Vertical Shift: $d$

Example: Graph $y = \cot(2x)$

• Period: $\frac{\pi}{2}$

• Vertical asymptotes at $x = 0, \frac{\pi}{2}, \pi, ...$

Practice Problem

Given the graph of $y = \cot(bx + c)$, determine the value of $b$ by analyzing the period and location of asymptotes.

• If the period is $\frac{\pi}{4}$, then $b = 4$.

Summary Table: Properties of Tangent and Cotangent Functions

Property	$\tan(x)$	$\cot(x)$
Definition	$\frac{\sin(x)}{\cos(x)}$	$\frac{\cos(x)}{\sin(x)}$
Period	$\pi$	$\pi$
Asymptotes	$x = \frac{\pi}{2} + k\pi$	$x = k\pi$
Zeros	$x = k\pi$	$x = \frac{\pi}{2} + k\pi$

Additional info:

• Graphs and examples illustrate how to sketch tangent and cotangent functions with various transformations.

• Practice problems reinforce understanding of period and asymptote calculation for transformed functions.