UNIT 9.6 THE TANGENT UNIT 9.6 THE TANGENT FUNCTIONFUNCTION

Warm Up

If sin A = , evaluate.

1. cos A 2. tan A

3. cot A 4. sec A

5. csc A

Recognize and graph trigonometric functions.

Objective

The tangent and cotangent functions can be graphed on the coordinate plane. The tangent function is undefined when θ = + πn, where n is an integer. The cotangent function is undefined when θ = πn. These values are excluded from the domain and are represented by vertical asymptotes on the graph. Because tangent and cotangent have no maximum or minimum values, amplitude is undefined.

To graph tangent and cotangent, let the variable x represent the angle θ in standard position.

Like sine and cosine, you can transform the tangent function.

Example 1: Transforming Tangent Functions

Step 2 Identify the x-intercepts.

The first x-intercept occurs at x = 0. Because the period is 3π, the x-intercepts occurs at 3πn where n is an integer.

Step 1 Identify the period.

Because b = the period is

Using f(x) = tan x as a guide, graph

Identify the period,

x-intercepts, and asymptotes.

g(x) =

Example 1 Continued

Step 3 Identify the asymptotes.

Step 4 Graph using all of the information about the function.

Because b = , the asymptotes occur at

Check It Out! Example 1

Step 1 Identify the period.

Step 2 Identify the x-intercepts.

The first x-intercept occurs at x = 0. Because the period is 2π, the x-intercepts occur at 2πn where n is an integer.

Because b = the period is

Using f(x) = tan x as a guide, graph .

Identify the period, x-intercepts, and asymptotes.

Check It Out! Example 1 Continued

Step 3 Identify the asymptotes.

Step 4 Graph using all of the information about the function.

Example 2: Graphing the Cotangent Function

Using f(x) = cot x as a guide, graph . Identify the period, x-intercepts, and asymptotes.

Step 1 Identify the period.

Because b = 3 the period is

Step 2 Identify the x-intercepts.

The first x-intercept occurs at x = . Because the period is , the x-intercepts occurs at

, where n is an integer.

Example 2: Graphing the Cotangent Function

Step 3 Identify the asymptotes.

Step 4 Graph using all of the information about the function.

Because b = 3, the asymptotes occur at

Check It Out! Example 2

Using f(x) = cot x as a guide, graph g(x) = –cot2x. Identify the period, x-intercepts, and asymptotes.

Step 1 Identify the period.

Because b = 2 the period is .

Step 2 Identify the x-intercepts.

The first x-intercept occurs at x = . Because the period is , the x-intercepts occurs at

, where n is an integer.

Check It Out! Example 2 Continued

Step 3 Identify the asymptotes.

Because b = 2, the asymptotes occur at

x =

Step 4 Graph using all of the information about the function.

Recall that sec θ = . So, secant is undefined where cosine equals zero and the graph will have vertical asymptotes at those locations. Secant will also have the same period as cosine. Sine and cosecant have a similar relationship. Because secant and cosecant have no absolute maxima, no minima, amplitude is undefined.

You can graph transformations of secant and cosecant by using what you learned in Lesson 14-1 about transformations of graphs of cosine and sine.

Example 3: Graphing Secant and Cosecant FunctionsUsing f(x) = cos x = as a guide, graph

g(x) = Identify the period and

asymptotes.Step 1 Identify the period.

Because sec is the reciprocal of

cos the graphs will have the same period.Because b = for cos the period is

Example 3 Continued

Step 2 Identify the asymptotes.

Step 3 Graph using all of the information about the function.

Because the period is 4π, the asymptotes

occur at

where n is an integer.

Check It Out! Example 3

Using f(x) = sin x as a guide, graph g(x) = 2csc x. Identify the period and asymptotes.

Step 1 Identify the period.

Because csc x is the reciprocal of

sin x the graphs will have the same period.

Because b = 1 for csc x the period is

Check It Out! Example 3 Continued

Step 2 Identify the asymptotes.

Because the period is 2π, the asymptotes

occur at

Step 3 Graph using all of the information about the function.

Lesson Quiz: Part I

1. Using f(x) = tan x as a guide, graph g(x) =

Identify the period, x-intercepts, and asymptotes.

period: 2π; x-intercepts: 2πn; asymptotes:

x = π + 2πn

Lesson Quiz: Part II

2. Using f(x) = sin(x) as a guide, graph g(x) =

Identify the period, and asymptotes.

period: 6π; asymptotes: x = 3πn

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