holt mcdougal algebra 2 graphs of other trigonometric functions holt algebra 2 warm up warm up...
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Holt McDougal Algebra 2
Graphs of Other Trigonometric Functions
Graphs of Other Trigonometric Functions
Holt Algebra 2
Warm UpWarm Up
Lesson PresentationLesson Presentation
Lesson QuizLesson Quiz
Holt McDougal Algebra 2
Holt McDougal Algebra 2
Graphs of Other Trigonometric Functions
Warm Up
If sin A = , evaluate.
1. cos A 2. tan A
3. cot A 4. sec A
5. csc A
Holt McDougal Algebra 2
Graphs of Other Trigonometric Functions
Recognize and graph trigonometric functions.
Objective
Holt McDougal Algebra 2
Graphs of Other Trigonometric Functions
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.
Holt McDougal Algebra 2
Graphs of Other Trigonometric Functions
Holt McDougal Algebra 2
Graphs of Other Trigonometric Functions
Like sine and cosine, you can transform the tangent function.
Holt McDougal Algebra 2
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) =
Holt McDougal Algebra 2
Graphs of Other Trigonometric Functions
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
Holt McDougal Algebra 2
Graphs of Other Trigonometric Functions
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 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.
Holt McDougal Algebra 2
Graphs of Other Trigonometric Functions
Check It Out! Example 1 Continued
Step 3 Identify the asymptotes.
Step 4 Graph using all of the information about the function.
Holt McDougal Algebra 2
Graphs of Other Trigonometric Functions
Holt McDougal Algebra 2
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.
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.
Holt McDougal Algebra 2
Graphs of Other Trigonometric FunctionsExample 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
Holt McDougal Algebra 2
Graphs of Other Trigonometric Functions
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.
Holt McDougal Algebra 2
Graphs of Other Trigonometric Functions
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.
Holt McDougal Algebra 2
Graphs of Other Trigonometric Functions
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.
Holt McDougal Algebra 2
Graphs of Other Trigonometric Functions
Holt McDougal Algebra 2
Graphs of Other Trigonometric Functions
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.
Holt McDougal Algebra 2
Graphs of Other Trigonometric Functions
Example 3: Graphing Secant and Cosecant Functions
Using 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
Holt McDougal Algebra 2
Graphs of Other Trigonometric Functions
Example 3 Continued
Step 2 Identify the asymptotes.
Step 3 Graph using all of the information about the function.
Because the period is 4, the asymptotes
occur at
where n is an integer.
Holt McDougal Algebra 2
Graphs of Other Trigonometric Functions
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
Holt McDougal Algebra 2
Graphs of Other Trigonometric Functions
Check It Out! Example 3 Continued
Step 2 Identify the asymptotes.
Because the period is 2, the asymptotes
occur at
Step 3 Graph using all of the information about the function.
Holt McDougal Algebra 2
Graphs of Other Trigonometric Functions
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
Holt McDougal Algebra 2
Graphs of Other Trigonometric Functions
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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