Graphs
GraphsofofOther
OtherTrigonometric
Trigonometric
11-2Functions
11-2
Functions
Warm Up
Lesson Presentation
Lesson Quiz
HoltMcDougal
Algebra 2Algebra 2
Holt
Graphs of Other Trigonometric
11-2 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
11-2 Functions
Objective
Recognize and graph trigonometric
functions.
Holt McDougal Algebra 2
Graphs of Other Trigonometric
11-2 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
11-2 Functions
Holt McDougal Algebra 2
Graphs of Other Trigonometric
11-2 Functions
Like sine and cosine, you can transform the
tangent function.
Holt McDougal Algebra 2
Graphs of Other Trigonometric
11-2 Functions
Example 1: Transforming Tangent Functions
Using f(x) = tan x as a guide, graph
g(x) =
Identify the period,
x-intercepts, and asymptotes.
Step 1 Identify the period.
Because b =
the period is
Step 2 Identify the x-intercepts.
The first x-intercept occurs at x = 0.
Because the period is 3, the x-intercepts
occurs at 3n where n is an integer.
Holt McDougal Algebra 2
Graphs of Other Trigonometric
11-2 Functions
Example 1 Continued
Step 3 Identify the asymptotes.
Because b =
, the asymptotes occur at
Step 4 Graph using all of the
information about
the function.
Holt McDougal Algebra 2
Graphs of Other Trigonometric
11-2 Functions
Check It Out! Example 1
Using f(x) = tan x as a guide, graph
.
Identify the period, x-intercepts, and asymptotes.
Step 1 Identify the period.
Because b =
the period is
Step 2 Identify the x-intercepts.
The first x-intercept occurs at x = 0. Because
the period is 2, the x-intercepts occur at 2n
where n is an integer.
Holt McDougal Algebra 2
Graphs of Other Trigonometric
11-2 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
11-2 Functions
Holt McDougal Algebra 2
Graphs of Other Trigonometric
11-2 Functions
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.
Holt McDougal Algebra 2
Graphs of Other Trigonometric
11-2 Functions
Example 2: Graphing the Cotangent Function
Step 3 Identify the asymptotes.
Because b = 3, the asymptotes occur at
Step 4 Graph using all of the
information about the
function.
Holt McDougal Algebra 2
Graphs of Other Trigonometric
11-2 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
11-2 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
11-2 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
11-2 Functions
Holt McDougal Algebra 2
Graphs of Other Trigonometric
11-2 Functions
You can graph transformations of secant and
cosecant by using what you learned in Lesson
11-1 about transformations of graphs of cosine
and sine.
Holt McDougal Algebra 2
Graphs of Other Trigonometric
11-2 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 =
2
Holt McDougal Algebra 2
for cos
the period is
Graphs of Other Trigonometric
11-2 Functions
Example 3 Continued
Step 2 Identify the asymptotes.
Because the period is 4, the asymptotes
occur at
where n is an integer.
Step 3 Graph using all of the
information about the
function.
Holt McDougal Algebra 2
Graphs of Other Trigonometric
11-2 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
2
Holt McDougal Algebra 2
Graphs of Other Trigonometric
11-2 Functions
Check It Out! Example 3 Continued
Step 2 Identify the asymptotes.
Step 3 Graph using all of
the information
about the function.
Holt McDougal Algebra 2
Graphs of Other Trigonometric
11-2 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: 2n; asymptotes:
x =  + 2n
Holt McDougal Algebra 2
Graphs of Other Trigonometric
11-2 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 = 3n
Holt McDougal Algebra 2