Graphs of Secant and
Cosecant
Section 4.5b
The graph of the secant function
1
y  sec x 
cos x
Wherever cos(x) = 1, its reciprocal sec(x) is also 1.
The graph has asymptotes at the zeros of the
cosine function.
The period of the secant function is
as the cosine function.
2 , the same
A local maximum of y = cos(x) corresponds to a
local minimum of y = sec(x), and vice versa.
The graph of the secant function
1
y  sec x 
cos x
1
2

1

2
The graph of the cosecant function
1
y  csc x 
sin x
Wherever sin(x) = 1, its reciprocal csc(x) is also 1.
The graph has asymptotes at the zeros of the sine
function.
The period of the cosecant function is
same as the sine function.
2 , the
A local maximum of y = sin(x) corresponds to a
local minimum of y = csc(x), and vice versa.
The graph of the cosecant function
1
y  csc x 
sin x
1
2

1

2
Summary: Basic Trigonometric Functions
Function
Period
Domain
sin x
2
cos x
2
 ,  
 ,  
tan x
cot x


x   2  n
x  n
sec x
2
x   2  n
csc x
2
x  n
Range
1,1
1,1
 ,  
 ,  
 , 1 1,  
 , 1 1,  
Summary: Basic Trigonometric Functions
Function
Asymptotes
Zeros
Even/Odd
sin x
None
n
Odd
cos x
None
 2  n
Even
cot x
x   2  n n
x  n
 2  n
sec x
x   2  n
None
Even
csc x
x  n
None
Odd
tan x
Odd
Odd
Guided Practice
Solve for x in the given interval  No calculator!!!
sec x  2
3
 x
2
Let’s construct a reference triangle:
240
–1
 Third Quadrant
r
sec x  2 
x
r  2, x  1
Convert to radians:
60
2
4
x  240 
3
Whiteboard Problem
Solve for x in the given interval  No calculator!!!
3
sec x   2   x 
2
5
x
4
Whiteboard Problem
Solve for x in the given interval  No calculator!!!
3
x
cot x  1   x  
2
4

Guided Practice
Use a calculator to solve for x in the given interval.
3
csc x  1.5   x 
2
The reference triangle:
x
1
 Third Quadrant
r
csc x  1.5 
y
r  1.5, y  1
1
sin x  
1.5
1.5
x    sin
1
 2 3  3.871
Does this answer make sense with our graph?
Guided Practice
Use a calculator to solve for x in the given interval.
tan x  0.3 0  x  2
Possible reference triangles:
y
tan x  0.3 
x
y  0.3, x  1
-1
x
x
1
-0.3
0.3
x  tan
1
 0.3  0.291
or
x    tan
1
 0.3  3.433