Trigonometry Lecture Notes
Section 4.4
Page 1 of 6
Section 4.4: Graphs of the Secant and Cosecant Functions
Big Idea: Learning the shape of these graphs allows for quick sketching of linearly transformed
graphs.
Big Skill: You should be able to sketch the secant and cosecant graphs quickly, including their
period and locations of extrema and zeros, and also sketch linear transformations of these
functions.
Trigonometry Lecture Notes
Section 4.4
Page 2 of 6
Graph of the Secant Function
1
y  f  x   sec  x  
cos  x 
Domain:
Range:
Table of Values:
y  sec  x 
x
0

6

4

3

2
2
3
3
4
5
6

5
6
5
4
4
3
3
2
5
3
7
4
11
6
Notes on the graph of the secant function:
 The period is 2.
 2n  1 

The graph is discontinuous at x-values



multiples of 2 ).
There are no x-intercepts.
The graph has no amplitude, since it goes to .
The graph has “local” extrema of 1 at integer multiples of   sec  x   1

The secant function is an even function  sec   x   sec  x  .
2
, where n is an integer (i.e., at odd
Trigonometry Lecture Notes
Section 4.4
Page 3 of 6
Graph of the Cosecant Function
1
y  f  x   csc  x  
sin  x 
Domain:
Range:
Table of Values:
y  f  x   csc  x 
x
0

6

4

3

2
2
3
3
4
5
6

5
6
5
4
4
3
3
2
5
3
7
4
11
6
Notes on the graph of the cosecant function:
 The period is 2.
 The graph is discontinuous at integer multiples of .
 There are no x-intercepts.
 The graph has no amplitude, since it goes to .
 The graph has “local” extrema of 1 at odd multiples of


2
 csc  x   1
The cosecant function is an odd function  csc   x    csc  x  .
Trigonometry Lecture Notes
Section 4.4
Page 4 of 6
Note that since the secant function is the reciprocal of the cosine function, sec  x  
1
, the
cos  x 
graphs of both functions intersect where the output is 1, and the secant function blows up where
the cosine function goes to zero:
Also note that since the cosecant function is the reciprocal of the sine function, csc  x  
1
,
sin  x 
the graphs of both functions intersect where the output is 1, and the cosecant function blows up
where the sine function goes to zero:
Trigonometry Lecture Notes
Section 4.4
Practice:
1. Graph the equation y  3sec  2 x  .
2. Graph the equation y  12 csc  x  4  .
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Trigonometry Lecture Notes
Section 4.4
3. Graph the equation y  sin  x   cos  x  .
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