Tangent, Cotangent, Secant, and Cosecant Functions in Precalculus

Trigonometric Functions: Tangent, Cotangent, Secant, and Cosecant

Overview

This section explores the remaining four trigonometric functions—tangent, cotangent, secant, and cosecant—their properties, graphs, and applications. Understanding these functions is essential for analyzing periodic phenomena and solving trigonometric equations.

The Tangent Function

Definition and Properties

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

• Domain: All real numbers except odd multiples of $\frac{\pi}{2}$, where $\cos x = 0$.

• Range: $(-\infty, \infty)$ (not bounded above or below).

• Periodicity: Period of $\pi$.

• Symmetry: Odd function; symmetric with respect to the origin.

• Continuity: Continuous on its domain; discontinuous at vertical asymptotes.

• Asymptotes: Vertical asymptotes at $x = (2k+1)\frac{\pi}{2}$ for all integers $k$.

• Zeros: At integer multiples of $\pi$ (where $\sin x = 0$).

• Monotonicity: Increasing on each interval of its domain.

• Inflection Points: At all integer multiples of $\pi$.

Example: Graphing a Transformed Tangent Function

• Consider $y = 3 \tan\left(\frac{x}{2}\right) + 1$.

• Transformations:

  • Horizontal stretch by a factor of 2

  • Vertical stretch by a factor of 3

  • Vertical translation up 1 unit

• Period: $\frac{\pi}{b}=\frac{\pi}{\frac12}=2\pi$

• Vertical Asymptotes: At even multiples of $\pi$

The Cotangent Function

Definition and Properties

• Definition: $\cot x = \frac{\cos x}{\sin x}$ (reciprocal of tangent).

• Domain: All real numbers except integer multiples of $\pi$ (where $\sin x = 0$).

• Range: $(-\infty, \infty)$.

• Periodicity: Period of $\pi$.

• Asymptotes: Vertical asymptotes at $x = k\pi$ for all integers $k$.

• Zeros: At odd multiples of $\frac{\pi}{2}$ (where $\cos x = 0$).

The Secant Function

Definition and Properties

• Definition: $\sec x = \frac{1}{\cos x}$ (reciprocal of cosine).

• Domain: All real numbers except odd multiples of $\frac{\pi}{2}$ (where $\cos x = 0$).

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

• Periodicity: Period of $2\pi$.

• Asymptotes: Vertical asymptotes at $x = (2k+1)\frac{\pi}{2}$ for all integers $k$.

The Cosecant Function

Definition and Properties

• Definition: $\csc x = \frac{1}{\sin x}$ (reciprocal of sine).

• Domain: All real numbers except integer multiples of $\pi$ (where $\sin x = 0$).

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

• Periodicity: Period of $2\pi$.

• Asymptotes: Vertical asymptotes at $x = k\pi$ for all integers $k$.

Solving Trigonometric Equations Graphically

Example: Intersection of Trigonometric Graphs

• To solve equations such as $\sec x = \csc x$ graphically, plot both functions and find their intersection points.

• Using a graphing calculator, the smallest positive solution is $x \approx 0.7749$.

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

Function	Definition	Domain	Range	Period	Vertical Asymptotes	Zeros
$\tan x$	$\frac{\sin x}{\cos x}$	$x \neq (2k+1)\frac{\pi}{2}$	$(-\infty, \infty)$	$\pi$	$x = (2k+1)\frac{\pi}{2}$	$x = k\pi$
$\cot x$	$\frac{\cos x}{\sin x}$	$x \neq k\pi$	$(-\infty, \infty)$	$\pi$	$x = k\pi$	$x = (2k+1)\frac{\pi}{2}$
$\sec x$	$\frac{1}{\cos x}$	$x \neq (2k+1)\frac{\pi}{2}$	$(-\infty, -1] \cup [1, \infty)$	$2\pi$	$x = (2k+1)\frac{\pi}{2}$	None
$\csc x$	$\frac{1}{\sin x}$	$x \neq k\pi$	$(-\infty, -1] \cup [1, \infty)$	$2\pi$	$x = k\pi$	None

Additional info: The above table summarizes the main properties of the four trigonometric functions discussed in this section, providing a quick reference for their definitions, domains, ranges, periods, asymptotes, and zeros.