Transcript ppt

Objectives:
Graphs of Other Trigonometric Functions
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Understand the graph of y = tan x.
Graph variations of y = tan x.
Understand the graph of y = cot x.
Graph variations of y = cot x.
Understand the graphs of y = csc x and y = sec x.
Graph variations of y = csc x and y = sec x.
Dr .Hayk Melikyan
Department of Mathematics and CS
[email protected]
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The Graph of y = tan x
Period: 
The tangent function is an odd function.
tan(  x )   tan x
The tangent function is undefined at
x

.
2
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The Tangent Curve: The Graph of y = tan x and Its
Characteristics
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3
The Tangent Curve: The Graph of y = tan x and Its
Characteristics (continued)
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Graphing Variations of y = tan x
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Graphing Variations of y = tan x
(continued)
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Example: Graphing a Tangent Function
Graph y = 3 tan 2x for    x  3 .
4
4
A = 3, B = 2, C = 0
Step 1 Find two consecutive asymptotes.


2
 Bx  C 

2


2
 2x 


2

 x

4
4
An interval containing one period is    ,   . Thus, two

4 4
consecutive asymptotes occur at x   
4
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and x   .
4
7
Example: Graphing a Tangent Function
(continued)
Graph y = 3 tan 2x for    x  3 .
4
4
Step 2 Identify an x-intercept, midway between the
consecutive asymptotes.


x = 0 is midway between  and .
4
4
The graph passes through (0, 0).
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Example: Graphing a Tangent Function
(continued)
Graph y = 3 tan 2x for    x  3 .
4
4
Step 3 Find points on the graph 1/4 and 3/4 of the way
between the consecutive asymptotes. These points
have y-coordinates of –A and A.
 3  3 tan 2 x
 1  tan 2 x
  
2x    
 4
x

8
3  3 tan 2 x
1  tan 2 x
 
2x   
4

x
8
The graph passes through
 

 
  ,  3  and  , 3  .
 8

8 
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Example: Graphing a Tangent Function
(continued)
Graph y = 3 tan 2x for    x  3 .
4
4
Step 4 Use steps 1-3 to
graph one full period
of the function.
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The Cotangent Curve: The Graph of y = cot x and Its
Characteristics
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The Cotangent Curve: The Graph of y = cot x and Its
Characteristics (continued)
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Graphing Variations of y = cot x
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Graphing Variations of y = cot x
(continued)
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Example: Graphing a Cotangent Function
1

2
2
Graph y  cot
A
1
2
,B 

x
,C  0
2
Step 1 Find two consecutive asymptotes.
0  Bx  C  
0

x
0 x2
2
An interval containing one period is (0, 2). Thus, two
consecutive asymptotes occur at x = 0 and x = 2.
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15
Example: Graphing a Cotangent Function
1

2
2
Graph y  cot
(continued)
x
Step 2 Identify an x-intercept midway between the
consecutive asymptotes.
x = 1 is midway between x = 0 and x = 2.
The graph passes through (1, 0).
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Example: Graphing a Cotangent Function (continued)
1

2
2
Graph y  cot
x
Step 3 Find points on the graph 1/4 and 3/4 of the way
between consecutive asymptotes. These points have
y-coordinates of A and –A.

1

1
cot

x
 1  cot

2
2
2
2
1
1



cot
x
1  cot
2
x
3

4
x


x
x
2


2
x
x
4
2
1 1
3 1
The graph passes through  ,   and  ,  .
2 2
2 2
2
2
2
3
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1
2
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Example: Graphing a Cotangent Function
1

2
2
Graph y  cot
(continued)
x
Step 4 Use steps 1-3 to
graph one full period
of the function.
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The Graphs of y = csc x and y = sec x
We obtain the graphs of the cosecant and the secant curves by
using the reciprocal identities
csc x 
1
sin x
and
sec x 
1
.
cos x
We obtain the graph of y = csc x by taking reciprocals of the
y-values in the graph of y = sin x. Vertical asymptotes of
y = csc x occur at the x-intercepts of y = sin x.
We obtain the graph of y = sec x by taking reciprocals of the
y-values in the graph of y = cos x. Vertical asymptotes of
y = sec x occur at the x-intercepts of y = cos x.
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The Cosecant Curve: The Graph of y = csc x and Its
Characteristics
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The Cosecant Curve: The Graph of y = csc x and Its
Characteristics (continued)
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The Secant Curve: The Graph of y = sec x and Its
Characteristics
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The Secant Curve: The Graph of y = sec x and Its
Characteristics (continued)
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Example: Using a Sine Curve to Obtain a Cosecant Curve
Use the graph of


y  csc  x 
4

 

y  sin  x   to obtain the graph of
4


.

The x-intercepts of
the sine graph correspond
to the vertical asymptotes
of the cosecant graph.
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Example: Using a Sine Curve to Obtain a Cosecant Curve
(continued)
Use the graph of
 
 

y  sin  x   to obtain the graph of
4

 

y  csc  x  
4


y  csc  x   .
4

Using the asymptotes as guides,
we sketch the graph of


y  csc  x 
4


.

 

y  sin  x  
4

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Example: Graphing a Secant Function
Graph y = 2 sec 2x for 
3
4
 x
3
.
4
We begin by graphing the reciprocal function, y = 2 cos 2x.
This equation is of the form y = A cos Bx, with A = 2 and
B = 2.
amplitude:
period:
2
B
A  2 2

2

We will use quarter-periods
to find x-values for the
five key points.
2
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Example: Graphing a Secant Function
Graph y = 2 sec 2x for 
3
4
 x
3
(continued)
.
4
  3
The x-values for the five key points are: 0, , ,
4 2
, and  .
4
Evaluating the function y = 2 cos 2x at each of these values
of x, the key points are:

 
  3

(0, 2),  , 0  ,  ,  2  , 
, 0  , and   , 2  .
4  2
  4

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Example: Graphing a Secant Function
Graph y = 2 sec 2x for 
3
 x
4
3
(continued)
.
4
The key points for our graph of y = 2 cos 2x are:

 
  3

(0, 2),  , 0  ,  ,  2  , 
,0 ,
4  2
  4

and   , 2  .
We draw vertical asymptotes
through the x-intercepts to use
as guides for the graph of
y = 2 sec 2x.
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Example: Graphing a Secant Function
Graph y = 2 sec 2x for 
3
4
 x
3
(continued)
.
4
y  2 sec 2 x
y  2 cos 2 x
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The Six Curves of Trigonometry
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The Six Curves of Trigonometry (continued)
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The Six Curves of Trigonometry (continued)
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The Six Curves of Trigonometry (continued)
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The Six Curves of Trigonometry (continued)
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The Six Curves of Trigonometry (continued)
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