5.4 – Day 1
More Trigonometric Graphs
Objectives
► Graphs of Tangent, Cotangent, Secant, and
Cosecant
► Graphs of Transformation of Tangent and
Cotangent
► Graphs of Transformations of Cosecant and
Secant
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More Trigonometric Graphs
In this section, we graph the tangent, cotangent, secant,
and cosecant functions and transformations of these
functions.
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Graphs of Tangent and Cotangent
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Graphs of Tangent and Cotangent
We begin by stating the periodic properties of these
functions. Sine and cosine have period 2.
Since cosecant and secant are the reciprocals of sine and
cosine, respectively, they also have period 2. Tangent and
cotangent, however, have period .
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Graphs of Tangent and Cotangent
Let’s use the values from the unit circle and the function
y = tan x = sin x ÷ cos x to make a table of values and a
graph of the tangent function.
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Graphs of Tangent and Cotangent
The graph of y = tan x approaches the vertical lines
x =  /2 and x = – /2. So, these lines are vertical
asymptotes.
* Remember, the vertical asymptotes occur at the values
that make the denominator (cos x) equal to 0.
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Graphs of Tangent and Cotangent
With the information we have so far, we can sketch the
“standard” graph of y = tan x from – /2 < x <  /2.
One period of y = tan x
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Graphs of Tangent and Cotangent
The complete graph of tangent is now obtained using the
fact that tangent is periodic with period .
y = tan x
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Graphs of Tangent and Cotangent
Let’s use the values from the unit circle and the function
y = cot x = cos x ÷ sin x to make a table of values and a
graph of the cotangent function.
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Graphs of Tangent and Cotangent
The function y = cot x is graphed on the interval (0, ).
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Graphs of Tangent and Cotangent
Since cot x is undefined for x = n with n an integer, its
complete graph has vertical asymptotes at these values.
One period y = cot x
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Graphs of Transformations of
Tangent and Cotangent
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Graphs of Transformations of Tangent and Cotangent
We now consider graphs of transformations of the tangent
and cotangent functions.
Since the tangent and cotangent functions have period ,
the functions
y = a tan k(x-b)
and
y = a cot k(x-b)
(k > 0)
complete one period as kx varies from 0 to , that is, for
0  kx  . Solving this inequality, we get 0  x   /k. So,
they each have period  /k.
Remember, the value of “a” stretches or compresses the
graph vertically.
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Graphs of Transformations of Tangent and Cotangent
Thus, one complete period of the graphs of these functions
occurs on any interval of length  /k.
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Graphs of Transformations of Tangent and Cotangent
To sketch a complete period of these graphs, it’s
convenient to select an interval between vertical
asymptotes:
To find consecutive vertical asymptotes for the graph of
y = a tan k(x – b), solve the equations
k(x – b) = -π/2
and
k(x – b) = π/2
To find consecutive vertical asymptotes for the graph of
y = a cot k(x – b), solve the equations
k(x – b) = 0
and
k(x – b) = π
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Example 2 – Graphing Tangent Curves
Graph the function:
(a) y = tan 2x
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Example 2 – Graphing Tangent Curves
Graph the function:
(b) y = tan 2
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Example 2 – Graphing Tangent Curves
(a) y = tan 2x
(b) y = tan 2
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Example 3 – A Shifted Cotangent Curve
Graph y = 2 cot
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Example 3 – A Shifted Cotangent Curve
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Graphs of Cosecant and Secant
It is apparent that the graphs of y = tan x and y = cot x are
symmetric about the origin.
This is because tangent and cotangent are odd functions.
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More Trigonometric Graphs
Practice:
p. 405-406
#1, 3, 5, 6, 11, 23, 35, 43, 53, 57
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5.4 – Day 2
More Trigonometric Graphs
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Objectives
► Graphs of Tangent, Cotangent, Secant, and
Cosecant
► Graphs of Transformation of Tangent and
Cotangent
► Graphs of Transformations of Cosecant and
Secant
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Graphs of Cosecant and Secant
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Graphs of Cosecant and Secant
Recall that since cosecant and secant are the reciprocals
of sine and cosine, respectively, they also have period 2.
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Graphs of Cosecant and Secant
To graph the cosecant and secant functions, we use the
reciprocal identities:
and
To graph y = csc x, we take the reciprocals of the
y-coordinates of the points of the graph of y = sin x.
So, let’s do that!
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Graphs of Cosecant and Secant
Graph y = csc x.
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Graphs of Cosecant and Secant
One period of y = csc x
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Graphs of Cosecant and Secant
The complete graph is obtained from the fact that the
function cosecant is periodic with period 2.
Note that the graph has vertical asymptotes at the points
where sin x = 0, that is, at x = n, for n an integer.
y = csc x
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Graphs of Cosecant and Secant
Similarly, to graph y = sec x, we take the reciprocals of the
y-coordinates of the points of the graph of y = cos x.
So, let’s do that!
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Graphs of Cosecant and Secant
Graph y = sec x.
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Graphs of Cosecant and Secant
One period of y = sec x
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Graphs of Cosecant and Secant
The complete graph of y = sec x is sketched in a similar
manner. Observe that the domain of sec x is the set of all
real numbers other than x = ( /2) + n, for n an integer, so
the graph has vertical asymptotes at those points.
y = sec x
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Graphs of Transformations of
Cosecant and Secant
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Graphs of Transformations of Cosecant and Secant
An appropriate interval on which to graph one complete
period is [0, 2 /k].
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Graphs of Transformations of Cosecant and Secant
To find consecutive vertical asymptotes for the graph of
y = a csc k(x – b), solve the equations
k(x – b)= 0
and
k(x – b)= π
To find consecutive vertical asymptotes for the graph of
y = a sec k(x – b), solve the equations
k(x – b)= -π/2
and
k(x – b)= π/2
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Example 4 – Graphing Cosecant Curves
Graph the function:
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Example 4 – Graphing Cosecant Curves
Graph the function:
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Example 4 – Graphing Cosecant Curves
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Example 5 – Graphing a Secant Curve
Graph y = 3 sec
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Example 5 – Graphing a Secant Curve
y = 3 sec
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Graphs of Cosecant and Secant
It is apparent that the graph of y = csc x is symmetric about
the origin, whereas that of y = sec x is symmetric about the
y-axis.
This is because cosecant is an odd function, whereas
secant is an even function.
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More Trigonometric Graphs
Practice:
p. 405
#2, 4, 7, 8, 15, 17, 21, 25, 31, 33,
45, 51
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