Holt McDougal Algebra 2 Graphs of Other Trigonometric Functions Holt Algebra 2 Warm Up Warm Up Lesson Presentation Lesson Presentation Lesson Quiz Lesson

Holt McDougal Algebra 2

Graphs of Other Trigonometric Functions


Holt Algebra 2

Warm UpWarm Up

Lesson PresentationLesson Presentation

Lesson QuizLesson Quiz




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 = + n, where n is an integer. The cotangent function is undefined when θ = n. 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.


Graphs of Other Trigonometric FunctionsExample 1: Transforming Tangent Functions

Step 2 Identify the x-intercepts.

The first x-intercept occurs at x = 0. Because the period is 3, the x-intercepts occurs at 3n 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



The first x-intercept occurs at x = 0. Because the period is 2, the x-intercepts occur at 2n 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






Graphs of Other Trigonometric FunctionsExample 2: Graphing the Cotangent Function

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


Because b = 3 the period is


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

, where n is an integer.


Graphs of Other Trigonometric FunctionsExample 2: Graphing the Cotangent Function



Because b = 3, the asymptotes occur at




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


Because b = 2 the period is .


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

, where n is an integer.





Because b = 2, the asymptotes occur at

x =




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 Functions

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

g(x) = Identify the period and

asymptotes.


Because sec is the reciprocal of

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



Example 3 Continued



Because the period is 4, the asymptotes

occur at

where n is an integer.




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


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





Because the period is 2, the asymptotes

occur at




Lesson Quiz: Part I

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

Identify the period, x-intercepts, and asymptotes.

period: 2; x-intercepts: 2n; asymptotes:

x = + 2n



Lesson Quiz: Part II

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

Identify the period, and asymptotes.

period: 6; asymptotes: x = 3n

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Holt McDougal Algebra 2 Graphs of Other Trigonometric Functions Holt Algebra 2 Warm Up Warm Up Lesson Presentation Lesson Presentation Lesson Quiz Lesson