1. No category

Objectives:
Graphs of Other Trigonometric Functions
•
•
•
•
•
•
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]
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
1
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
.
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
2
The Tangent Curve: The Graph of y = tan x and Its
Characteristics
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
3
The Tangent Curve: The Graph of y = tan x and Its
Characteristics (continued)
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
4
Graphing Variations of y = tan x
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
5
Graphing Variations of y = tan x
(continued)
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
6
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


4
x

4
An interval containing one period is    ,   . Thus, two
 4 4
consecutive asymptotes occur at x    and x   .
4
4
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
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).
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
8
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  3tan 2x
3  3tan 2x
The graph passes through
1  tan 2x
1  tan 2x
   , 3  and   ,3  .








2x    
2x   
 8

8 
 4
4


x
x
8
8
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
9
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.
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
10
The Cotangent Curve: The Graph of y = cot x and Its
Characteristics
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
11
The Cotangent Curve: The Graph of y = cot x and Its
Characteristics (continued)
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
12
Graphing Variations of y = cot x
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
13
Graphing Variations of y = cot x
(continued)
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
14
Example: Graphing a Cotangent Function
1 
Graph y  cot x
2
2
1

A  , B  ,C  0
2
2
Step 1 Find two consecutive asymptotes.
0  Bx  C  
0

2
x 
0 x2
An interval containing one period is (0, 2). Thus, two
consecutive asymptotes occur at x = 0 and x = 2.
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
15
Example: Graphing a Cotangent Function
(continued)
1 
Graph y  cot x
2
2
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).
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
16
Example: Graphing a Cotangent Function (continued)
1 
Graph y  cot x
2
2
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 

3 
3
  cot x 1  cot x
 x x
2 2
2
2
4 2
2

1
 
1 1 
1

cot
x
x
 x
 cot x
2
2
4 2
2 2
2
 1 , 1 .
 3, 1 
The graph passes through 

 and 
2 2
2 2
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
17
Example: Graphing a Cotangent Function
(continued)
1 
Graph y  cot x
2
2
Step 4 Use steps 1-3 to
graph one full period
of the function.
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
18
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
1
csc x 
sin x
and
1
sec x 
.
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.
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
19
The Cosecant Curve: The Graph of y = csc x and Its
Characteristics
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
20
The Cosecant Curve: The Graph of y = csc x and Its
Characteristics (continued)
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
21
The Secant Curve: The Graph of y = sec x and Its
Characteristics
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
22
The Secant Curve: The Graph of y = sec x and Its
Characteristics (continued)
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
23
Example: Using a Sine Curve to Obtain a Cosecant Curve


Use the graph of y  sin  x   to obtain the graph of
4



y  csc  x   .
4

The x-intercepts of
the sine graph correspond
to the vertical asymptotes
of the cosecant graph.
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
24
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  
y  csc  x   .
4

4

Using the asymptotes as guides,
we sketch the graph of

x 

y

sin
y  csc  x   .


4


4

Copyright © 2014, 2010, 2007 Pearson Education, Inc.
25
Example: Graphing a Secant Function
3
3
Graph y = 2 sec 2x for   x  .
4
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:
A  2 2
2

2

period:


B
2
We will use quarter-periods
to find x-values for the
five key points.
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
26
Example: Graphing a Secant Function
(continued)
3
3
Graph y = 2 sec 2x for   x  .
4
4
  3
The x-values for the five key points are: 0, , , , and  .
4 2 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 
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
27
Example: Graphing a Secant Function
(continued)
3
3
Graph y = 2 sec 2x for   x  .
4
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.
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
28
Example: Graphing a Secant Function
(continued)
3
3
Graph y = 2 sec 2x for   x  .
4
4
y  2sec 2 x
y  2cos 2 x
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
29
The Six Curves of Trigonometry
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
30
The Six Curves of Trigonometry (continued)
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
31
The Six Curves of Trigonometry (continued)
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
32
The Six Curves of Trigonometry (continued)
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
33
The Six Curves of Trigonometry (continued)
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
34
The Six Curves of Trigonometry (continued)
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
35