Graphs of Other Trigonometric Functions

Tangent Function: y = tan x

The tangent function is a fundamental trigonometric function with unique graph characteristics. It is defined as the ratio of sine to cosine: . The graph of y = tan x exhibits periodic behavior and vertical asymptotes where the function is undefined.

• Period:

• Domain: All real numbers except odd multiples of

• Range: All real numbers

• Vertical Asymptotes: At , where is an integer

• Odd Function: (origin symmetry)

• x-intercepts: Occur at , midway between consecutive asymptotes

• Key Points: At and of the way between asymptotes, y-values are and $1$ respectively

Example Table: Values of y = tan x

As x increases from 0 toward , tan x increases slowly at first, then more and more rapidly.

Graphing Variations of y = tan x

To graph , follow these steps:

• Find two consecutive asymptotes by solving

• Identify the x-intercept midway between the asymptotes

• Find points at and of the way between asymptotes, with y-coordinates and

• Graph one full period and extend as needed

Cotangent Function: y = cot x

The cotangent function is the reciprocal of the tangent function: . Its graph has similar properties but different locations for asymptotes and intercepts.

• Period:

• Domain: All real numbers except integral multiples of

• Range: All real numbers

• Vertical Asymptotes: At , where is an integer

• Odd Function: (origin symmetry)

• x-intercepts: Occur midway between consecutive asymptotes

• Key Points: At of the way between asymptotes, y-values are $1-1$ respectively

Graphing Variations of y = cot x

To graph , follow these steps:

• Find two consecutive asymptotes by solving

• Identify the x-intercept midway between the asymptotes

• Find points at and of the way between asymptotes, with y-coordinates and

• Graph one full period and extend as needed

Cosecant and Secant Functions: y = csc x and y = sec x

The cosecant and secant functions are the reciprocals of sine and cosine, respectively. Their graphs are constructed by taking the reciprocal of the y-values of the sine and cosine graphs, resulting in vertical asymptotes at the x-intercepts of the original functions.

• Cosecant:

• Secant:

• Period:

• Domain: All real numbers except where sine or cosine is zero

• Range:

• Vertical Asymptotes: At x-intercepts of sine (for csc) and cosine (for sec)

• Symmetry: Cosecant is odd, secant is even

Example: Constructing the Cosecant Curve from the Sine Curve

To graph , use the graph of and draw vertical asymptotes at its x-intercepts. The reciprocal values form the branches of the cosecant curve.

Example: Graphing a Secant Function

To graph , first graph , then use its x-intercepts as vertical asymptotes for the secant graph. The secant curve consists of branches above and below the cosine curve, where the reciprocal values are defined.

Summary Table: The Six Trigonometric Curves

Visual Summary of Trigonometric Curves