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1.6
GRAPHS OF OTHER TRIGONOMETRIC FUNCTIONS
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Graph of the Tangent Function
Recall that the tangent function is odd. That is,
tan(–x) = – tan x.
We also know from the identity tan x = sin x / cos x that the
tangent is undefined for values at which cos x = 0.
Two such values are x = ± / 2.
Graph of the Tangent Function
tan x increases without bound as x approaches  /2 from
the left,
and decreases without bound as x approaches – /2 from
the right.
Graph of the Tangent Function
The graph of y = tan x has vertical asymptotes at x =  /2
and x = – /2.
Graph of the Tangent Function
Since tangent has a period of , vertical asymptotes also occur
when x =  /2 + n, where n is an integer.
The domain is the set of all real numbers other than x =  /2 + n
The range is the set of all real numbers.
Graph of the Tangent Function
Two consecutive vertical asymptotes can be found by
solving the equations
bx – c =
and
bx – c =
The midpoint between two consecutive vertical asymptotes
is an x-intercept of the graph.
The period of the function y = a tan(bx – c) is the distance
between two consecutive vertical asymptotes.
Graph of the Tangent Function
After plotting the asymptotes and the x-intercept, plot a few
additional points between the two asymptotes and sketch
one cycle.
Example 1 – Sketching the Graph of a Tangent Function
Sketch the graph of y = tan(x / 2).
By solving the equations
x = –
x=
you can see that two consecutive vertical asymptotes occur
at x = – and x = .
Example 1 – Solution
cont’d
Between these two asymptotes, plot a few points, including
the x-intercept, as shown in the table.
Example 1 – Solution
cont’d
Graph of the Cotangent Function
Like tangent, the cotangent function has a period of .
The cotangent function has vertical asymptotes when sin x
is zero at
x = n, where n is an integer.
Graph of the Cotangent Function
Two consecutive vertical asymptotes of the graph of y = a
cot(bx – c) can be found by solving the equations
bx – c = 0 and bx – c = .
Example 3 – Sketching the Graph of a Cotangent Function
Sketch the graph of
By solving the equations
=0
x=0
=
x = 3
you can see that two consecutive vertical asymptotes occur
at x = 0 and x = 3 .
Example 3 – Solution
cont’d
Between these two asymptotes, plot a few points, including
the x-intercept.
Example 3 – Solution
The period is 3, the distance between consecutive
asymptotes.
cont’d
Graphs of the Reciprocal Functions
Using the sine and cosine we can graph the two remaining
trigonometric functions.
csc x =
1
sin 𝑥
and
sec x =
1
cos 𝑥
Graphs of the Reciprocal Functions
The graphs of
tan x =
and
sec x =
have vertical asymptotes at x =  /2 + n, and the cosine is
zero at these x-values. Similarly,
cot x =
and
csc x =
have vertical asymptotes where sin x = 0—that is, at x = n.
Graphs of the Reciprocal Functions
To sketch the graph of a secant or cosecant function, you
should first make a sketch of its reciprocal function.
For instance, to sketch the graph of y = csc x, first sketch
the graph of y = sin x.
Then draw the asymptotes, where sin x = 0.
Then, using parabolas, graph y = csc x above/below
y = sin x.
Graphs of the Reciprocal Functions
Graphs of the Reciprocal Functions
Use the sine and cosine as “guide graphs.”
Asymptotes (where the graph is
undefined) occur where sine/cosine
intersect the x-axis.
Example 4 – Sketching the Graph of a Cosecant Function
Sketch the graph of y = 2 csc
Begin by sketching the guide graph of
y = 2 sin
The amplitude is
2
The period is
2.
Example 4 – Solution
By solving the equations
x=
x=
We see that one cycle of the sine function corresponds to
the interval from x = – /4 to x = 7 /4.
cont’d
Since the period is 2π, we obtain the following interval
𝑝𝑒𝑟𝑖𝑜𝑑
interval =
4
2𝜋
𝜋
=
=
4
2
So, the five key points are
Key points
Intercept
x
y
−
𝜋
2
units apart
Maximum
Intercept
Minimum
Intercept
𝜋
4
𝜋
4
3𝜋
4
5𝜋
4
7𝜋
4
0
2
0
-2
0
Example 4 – Solution
Because the sine function is zero at the intercepts, the
cosecant graph has vertical asymptotes at
x = – /4, x = 3 /4, x = 7 /4
cont’d
Trigonometric Graphs
Trigonometric Graphs
cont’d