Graphs of the Other Trigonometric Functions

Section 4.5: Graphs of the Other Trigonometric Functions

Overview

This section explores the graphs of the other trigonometric functions: tangent, cotangent, secant, and cosecant. We will examine their shapes, asymptotes, domains, ranges, and key features, with examples to illustrate their properties.

The Tangent Function

Definition and Properties

The tangent function is defined as the ratio of the sine and cosine functions:

$\tan(x) = \frac{\sin(x)}{\cos(x)}, \quad x \neq \frac{\pi}{2} + n\pi, \ n \in \mathbb{Z}$

• Domain: All real numbers except $x = \frac{\pi}{2} + n\pi$, where $n$ is an integer (since $\cos(x) = 0$ at these points).

• Range: $(-\infty, \infty)$

• Period: $\pi$

• Vertical Asymptotes: $x = \frac{\pi}{2} + n\pi$

• Odd Function: $\tan(-x) = -\tan(x)$

Example: The graph of $y = \tan(x)$ repeats every $\pi$ units and has vertical asymptotes at $x = \frac{\pi}{2}, \frac{3\pi}{2}, \ldots$

The Cotangent Function

Definition and Properties

The cotangent function is the reciprocal of the tangent function, or the ratio of cosine to sine:

$\cot(x) = \frac{\cos(x)}{\sin(x)}, \quad x \neq n\pi, \ n \in \mathbb{Z}$

• Domain: All real numbers except $x = n\pi$, where $n$ is an integer (since $\sin(x) = 0$ at these points).

• Range: $(-\infty, \infty)$

• Period: $\pi$

• Vertical Asymptotes: $x = n\pi$

• Odd Function: $\cot(-x) = -\cot(x)$

Example: The graph of $y = \cot(x)$ repeats every $\pi$ units and has vertical asymptotes at $x = 0, \pi, 2\pi, \ldots$

Transformations of Tangent and Cotangent

Both tangent and cotangent functions can be transformed by changing their amplitude, period, phase shift, and vertical shift. For example, the function $y = a \tan(bx + c) + d$ has:

• Amplitude: Not defined for tangent/cotangent (since their range is all real numbers).

• Period: $\frac{\pi}{|b|}$

• Phase Shift: $-\frac{c}{b}$

• Vertical Shift: $d$

Example: $y = \cot 6\left(x - \frac{\pi}{6}\right)$ is a cotangent function with period $\frac{\pi}{6}$ and phase shift $\frac{\pi}{6}$ to the right.

The Cosecant Function

Definition and Properties

The cosecant function is the reciprocal of the sine function:

$\csc(x) = \frac{1}{\sin(x)}, \quad x \neq n\pi, \ n \in \mathbb{Z}$

• Domain: All real numbers except $x = n\pi$

• Range: $(-\infty, -1] \cup [1, \infty)$

• Period: $2\pi$

• Vertical Asymptotes: $x = n\pi$

Example: The graph of $y = \csc(x)$ has vertical asymptotes at $x = 0, \pi, 2\pi, \ldots$ and consists of branches above $y = 1$ and below $y = -1$.

Graphing Examples and Applications

Graphing Tangent and Cotangent with Transformations

When graphing functions such as $f(x) = \tan(3x + \frac{\pi}{2})$ or $f(x) = -3\cot(x - \frac{\pi}{3})$, identify the period, phase shift, and vertical asymptotes using the general forms:

• For $y = a \tan(bx + c) + d$, vertical asymptotes occur at $bx + c = \frac{\pi}{2} + n\pi$.

• For $y = a \cot(bx + c) + d$, vertical asymptotes occur at $bx + c = n\pi$.

Example: The graph of $f(x) = \tan(3x + \frac{\pi}{2})$ has a period of $\frac{\pi}{3}$ and is shifted horizontally.

Example: The graph of $f(x) = -3\cot(x - \frac{\pi}{3})$ has a period of $\pi$ and is reflected and shifted right by $\frac{\pi}{3}$.

Summary Table: Key Properties of Tangent, Cotangent, and Cosecant

Function	Definition	Domain	Range	Period	Vertical Asymptotes
$\tan(x)$	$\frac{\sin(x)}{\cos(x)}$	$x \neq \frac{\pi}{2} + n\pi$	$(-\infty, \infty)$	$\pi$	$x = \frac{\pi}{2} + n\pi$
$\cot(x)$	$\frac{\cos(x)}{\sin(x)}$	$x \neq n\pi$	$(-\infty, \infty)$	$\pi$	$x = n\pi$
$\csc(x)$	$\frac{1}{\sin(x)}$	$x \neq n\pi$	$(-\infty, -1] \cup [1, \infty)$	$2\pi$	$x = n\pi$