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Chapter 5
Trigonometric
Functions
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All rights reserved
© 2010
2011 Pearson Education, Inc. All rights reserved
1
SECTION 5.5
Graphs of the Other Trigonometric Functions
OBJECTIVES
1
2
Graph the tangent and cotangent functions.
Graph the cosecant and secant functions.
TANGENT FUNCTION
The tangent function differs from the sine and
cosine functions in three significant ways:
1. The tangent function has period π.
2. The tangent function is 0 when sin x = 0 and is
undefined when cos x = 0. It is undefined at
 3 5
 ,
,
, ...
2
2
2
3. The tangent function has no amplitude; that is,
there are no minimum and maximum y-values.
The range is (–∞, ∞).
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GRAPH OF THE TANGENT FUNCTION
Plotting y = tan x
for common values
of x and
connecting the
points with a
smooth curve
yields the
following:
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COTANGENT FUNCTION
The cotangent function is similar to the tangent
function:
1. The cotangent function has period π.
2. The cotangent function is 0 when cos x = 0 and
is undefined when sin x = 0. It is undefined at
 ,  3 ,  5 , ...
3. The cotangent function has no amplitude; that
is, there are no minimum and maximum
y-values. The range is (–∞, ∞).
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TANGENT AND COTANGENT FUNCTIONS
Both functions are odd:
tan (–x) = – tan x
cot (–x) = – cot x
1
1
cot x 
and tan x 
tan x
cot x
Both functions have the same sign everywhere
they are both defined.
When |tan x| is large, |cot x| is small (and
conversely).
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GRAPH OF THE COTANGENT FUNCTION
Using the
information on
the previous
two slides, we
graph y = cot x.
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MAIN FACTS ABOUT y = tan x AND y = cot x
y = tan x
Period
Domain
y = cot x
π
π
All real numbers All real numbers
except odd
except integer

multiples of aa
multiples of π
2
Range
Vertical
Asymptotes
(–∞,∞)
(–∞,∞)
x = odd

multiples of aa
x = integer
multiples of π
2
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MAIN FACTS ABOUT y = tan x AND y = cot x
x-intercepts
Symmetry
y = tan x
y = cot x
At integer
multiples of π
At odd multiples

of aa
2
tan (–x) = –tan x, cot (–x) = –cot x,
odd function;
odd function;
so symmetric
so symmetric
with respect to
with respect to
the origin
the origin
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EXAMPLE 2
Graphing
y = a cot [b(x – c)]


Graph y  4 cot  x   over the interval

2
[–π, 2π].
Solution


Step 1 For y  4 cot  x   , we have a = –4,

2
b = 1, and c 

2
.
Thus, vertical stretch factor = |–4| = 4
period 

1
 ;
phase shift 
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
2
10
EXAMPLE 2
Graphing
y = a cot [b(x – c)]
Solution continued
Step 2 Locate two adjacent asymptotes. Solve
x

2
0
x
and

2
x

 .
2
3
x
2
  3 
Step 3 The interval  ,  has length π.
2 2 
  3 
The division points of  ,  are:
2 2 
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EXAMPLE 2
Graphing
y = a cot [b(x – c)]
Solution continued
Step 3 continued
 1
3  1
   
,
     ,
2 4
4 2 2

3
5
   
.
and
2 4
4
Step 4 Evaluate the function at those points:
3
x
,
4
 3  
y  4cot 
   4
 4 2
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EXAMPLE 2
Graphing
y = a cot [b(x – c)]
Solution continued
Step 4 continued


x   , y  4cot      0
2

5
 5  
x
, y  4cot 
 4
4
 4 2
 3
Step 5 Sketch vertical asymptotes: x  ,
2 2
Draw one cycle through the points above.
Repeat the graph to the left and right
over intervals of π.
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EXAMPLE 2
Graphing
y = a cot [b(x – c)]
Solution continued


y  4cot  x  
2

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COSECANT FUNCTION
1
Cosecant is the reciprocal of sine: csc x 
sin x
Both functions have the same sign everywhere they
are both defined.
When |sin x| is large, |csc x| is small (and conversely).
csc x is undefined when sin x = 0. It is undefined at
0, ±π, ±2π, ±3π,…. At each of these points there is a
vertical asymptote.
csc x = 1 when sin x = 1 and csc x = –1 when
sin x = –1.
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GRAPH OF THE COSECANT FUNCTION
The graphs of
y = sin x and
y = csc x over
the interval
[–2π, 2π] are
shown to the
right.
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SECANT FUNCTION
1
Secant is the reciprocal of cosine: sec x 
cos x
Both functions have the same sign everywhere they
are both defined.
When |cos x| is large, |sec x| is small (and conversely).
sec x is undefined when cos x = 0. It is undefined at
 3 5
 ,  ,  , . At each of these points there is a
2
2
2
vertical asymptote.
sec x = 1 when cos x = 1 and sec x = –1 when
cos x = –1.
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GRAPH OF THE SECANT FUNCTION
The graphs
of y = cos x
and y = sec x
over the
interval
 3 5 
 2 , 2 
are shown to
the right.
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MAIN FACTS ABOUT y = csc x AND y = sec x
y = csc x
Period
Domain
y = sec x
2π
2π
All real numbers All real numbers
except integer
except odd

multiples of π
multiples of aa
2
Range
Vertical
Asymptotes
(–∞, –1] U [1, ∞) (–∞, –1] U [1, ∞)
x = integer
multiples of π
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x = odd multiples

of aa
2
19
MAIN FACTS ABOUT y = csc x AND y = sec x
x-intercepts
Symmetry
y = csc x
y = sec x
No x-intercepts
No x-intercepts
csc (–x) = –csc x, sec (–x) = sec x,
odd function,
even function
origin symmetry
with y-axis
symmetry
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EXAMPLE 4
Graphing a Range of Mach Numbers
When a plane travels at supersonic and hypersonic
speeds, small disturbances in the atmosphere are
transmitted downstream within a cone.
The cone intersects the ground, and the edge of the
cone’s intersection with the ground can be
represented as in the figure on the next slide.
The sound waves strike the edge of the cone at a
right angle. The speed of the sound wave is
represented by leg s of the right triangle shown in
the figure.
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EXAMPLE 4
Graphing a Range of Mach Numbers
The plane is
moving at speed
v, which is
represented by
the hypotenuse
of the right
triangle in
figure.
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EXAMPLE 4
Graphing a Range of Mach Numbers
The Mach number, M, is given by
speed of aircraft v
 x
M  M x  
  csc   ,
 2
speed of sound s
where x is the angle of the vertex of the cone.
Graph the Mach number function, M(x), as the
angle at the vertex of the cone varies. What is
the range of Mach numbers associated with the
 
interval  ,   ?
4 
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EXAMPLE 4
Graphing a Range of Mach Numbers
Solution
x
1
x
, first graph y  sin .
Because csc 
2 sin x
2
2
Then use the
reciprocal
connection.
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EXAMPLE 4
Graphing a Range of Mach Numbers
Solution continued

x

For x  , y  csc  csc  2.6.
4
2
8
x

For x   , y  csc  csc  1.
2
2
The range of Mach numbers associated with the
 
interval  ,   is (1,2.6].
4 
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