Trigonometric Functions: Graphs, Properties, and Transformations

Trigonometric Functions and Their Graphs

Overview of the Six Basic Trigonometric Functions

Trigonometric functions are fundamental in Precalculus, describing periodic phenomena and relationships in geometry. The six basic trigonometric functions are sine (sin), cosine (cos), tangent (tan), cosecant (csc), secant (sec), and cotangent (cot). Each function has unique properties, domains, ranges, and periodic behavior.

• Sine (sin) and cosine (cos) are periodic with period $2\pi$.

• Tangent (tan) and cotangent (cot) are periodic with period $\pi$.

• Cosecant (csc) and secant (sec) are reciprocals of sine and cosine, respectively.

Graphs of Reciprocal Trigonometric Functions

The reciprocal trigonometric functions—cosecant and secant—are defined as follows:

• $\csc(x) = \frac{1}{\sin(x)}$

• $\sec(x) = \frac{1}{\cos(x)}$

These functions have vertical asymptotes where their denominators are zero (i.e., where sine or cosine equals zero). Their graphs exhibit repeating patterns and are not defined at these points.

• Cosecant (csc): Period $2\pi$, asymptotes at $x = k\pi$ where $k$ is an integer.

• Secant (sec): Period $2\pi$, asymptotes at $x = \frac{\pi}{2} + k\pi$ where $k$ is an integer.

Graphs of Tangent and Cotangent Functions

The tangent and cotangent functions are defined as:

• $\tan(x) = \frac{\sin(x)}{\cos(x)}$

• $\cot(x) = \frac{\cos(x)}{\sin(x)}$

These functions have vertical asymptotes where their denominators are zero. Their graphs are periodic and pass through the origin (for tangent) or have undefined values at the origin (for cotangent).

• Tangent (tan): Period $\pi$, asymptotes at $x = \frac{\pi}{2} + k\pi$.

• Cotangent (cot): Period $\pi$, asymptotes at $x = k\pi$.

Summary Table: Properties of Trigonometric Functions

The following table summarizes the main properties of the six trigonometric functions:

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

Applications and Transformations

Understanding the graphs and properties of trigonometric functions is essential for solving equations, modeling periodic phenomena, and analyzing transformations such as shifts, stretches, and reflections. These concepts are foundational for further study in calculus and physics.