Graphs of Tangent, Cotangent, Cosecant, and Secant Functions in Precalculus 5.6

스터디 가이드 - 스마트 노트

자료에 맞춘 맞춤형 노트, 핵심 정의, 예시, 맥락을 확장해 제공합니다.

Trigonometric Functions

Graphs of the Tangent, Cotangent, Cosecant, and Secant Functions

This section explores the graphical behavior and properties of the tangent, cotangent, cosecant, and secant functions, which are fundamental in Precalculus. Understanding these graphs is essential for analyzing periodic phenomena and solving trigonometric equations.

Tangent Function

The tangent function, denoted as y = \tan x, is defined for all real numbers except where the cosine is zero (odd multiples of ). Its graph exhibits periodicity, vertical asymptotes, and symmetry about the origin.

Domain: All real numbers except , where is an integer.
Range: All real numbers.
Period:
Odd Function: Symmetric with respect to the origin.
x-intercepts: , is an integer.
y-intercept:
Vertical Asymptotes:

Example Table: Values of for selected :

x	y = tan x	(x, y)



$0$	$0$

	$1$

Table of values for y = tan x

Graph: The graph of between and shows the function's increasing behavior and vertical asymptotes.

Graph of y = tan x for -pi/3 to pi/3

Full Periodic Graph: The tangent function repeats every units, with vertical asymptotes at odd multiples of .

Periodic graph of y = tan x with vertical asymptotes

Cotangent Function

The cotangent function, denoted as y = \cot x, is defined for all real numbers except integer multiples of . Its graph is similar to tangent but shifted and reflected.

Domain: All real numbers except , is an integer.
Range: All real numbers.
Period:
Vertical Asymptotes:
x-intercepts:

Periodic graph of y = cot x with vertical asymptotes

Transformations of Tangent and Cotangent Functions

Functions of the form y = A \tan(\omega x) + B and y = A \cot(\omega x) + B involve amplitude, period, and vertical shifts. The period is affected by , and stretches or compresses vertically, while shifts the graph up or down.

Period: for tangent and cotangent.
Vertical Stretch/Compression: changes the steepness.
Vertical Shift: moves the graph up or down.

Example: Graph of shows vertical stretch and upward shift.

Graph of y = 3/2 tan x + 1/2 with transformations Graph of y = 3/2 tan x + 1/2 showing vertical shift

Example: Graphs of and illustrate vertical stretch and horizontal compression.

Graphs of cotangent transformations

Cosecant Function

The cosecant function, denoted as y = \csc x, is the reciprocal of the sine function. It is undefined where sine is zero (integer multiples of ), and its graph consists of branches separated by vertical asymptotes.

Domain: All real numbers except , is an integer.
Range:
Period:
Vertical Asymptotes:

Graph of y = csc x with vertical asymptotes

Secant Function

The secant function, denoted as y = \sec x, is the reciprocal of the cosine function. It is undefined where cosine is zero (odd multiples of ), and its graph consists of branches separated by vertical asymptotes.

Domain: All real numbers except , is an integer.
Range:
Period:
Vertical Asymptotes:

Graph of y = sec x with vertical asymptotes

Transformations of Cosecant and Secant Functions

Functions of the form y = A \csc(\omega x) + B and y = A \sec(\omega x) + B involve similar transformations as tangent and cotangent. The period is , stretches or compresses vertically, and shifts the graph.

Period:
Vertical Stretch/Compression: changes the height of the branches.
Vertical Shift: moves the graph up or down.

Example: Graphs of transformed cosecant and secant functions show how these changes affect the graph's shape and position.

Summary Table: Properties of Trigonometric Functions

Function	Domain	Range	Period	Vertical Asymptotes
		All real numbers
		All real numbers

Additional info: Transformations and properties are essential for solving trigonometric equations and modeling periodic phenomena in Precalculus.