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Graphs of Other
Trigonometric
Functions
The Tangent Curve: The Graph
of y=tanx and Its Characteristics
y
Period: 
Domain: All real numbers
except  /2 + k ,
k an integer
1
–2
–
5
2
–
–
3
2
–

2
–1

2
Range: All real numbers
2

0
3
2
5
2
x
Symmetric with respect to
the origin
Vertical asymptotes at
odd multiples of  /2
Graphing y = A tan(Bx – C)
y = A tan (Bx – C)
Bx – C = - /2
x-intercept
between
asymptotes
1. Find two consecutive asymptotes by setting the
variable expression in the tangent equal to -/2
and /2 and solving
Bx – C =  /2
Bx – C = -/2 and Bx – C = /2
2. Identify an x-intercept, midway between
consecutive asymptotes.
x
3. Find the points on the graph 1/4 and 3/4 of the
way between and x-intercept and the asymptotes.
These points have y-coordinates of –A and A.
4. Use steps 1-3 to graph one full period of the
function. Add additional cycles to the left or right
as needed.
Text Example
Graph y = 2 tan x/2 for – < x < 3 
Solution
Step 1 Find two consecutive asymptotes.
Thus, two consecutive asymptotes occur at x = -  and x = .
Step 2 Identify any x-intercepts, midway between consecutive
asymptotes. Midway between x = - and x =  is x = 0. An x-intercept is 0
and the graph passes through (0, 0).
Text Example cont.
Solution
Step 3 Find points on the graph 1/4 and 1/4 of the way between an xintercept and the asymptotes. These points have y-coordinates of –A and A.
Because A, the coefficient of the tangent, is 2, these points have y-coordinates of
-2 and 2.
Step 4 Use steps 1-3 to graph one
full period of the function. We use the
two consecutive asymptotes, x = - and
x = , an x-intercept of 0, and points
midway between the x-intercept and
asymptotes with y-coordinates of –2
and 2. We graph one full period of
y = 2 tan x/2 from – to . In order to
graph for – < x < 3 , we continue the
pattern and extend the graph another
full period on the right.
y = 2 tan x/2
y
4
2
˝
-˝
-2
-4
3˝
x
The Cotangent Curve: The Graph
of y = cotx and Its Characteristics
The Graph of y = cot x and Its Characteristics
Characteristics
y
4
2
-
-
 /2
 /2
-2
-4
˝
3
 /2
2
x
Period: 
Domain: All real numbers except
integral multiples of 
Range: All real numbers
Vertical asymptotes: at integral
multiples of 
n x-intercept occurs midway between
each pair of consecutive asymptotes.
Odd function with origin symmetry
Points on the graph 1/4 and 3/4 of the way
between consecutive asymptotes have ycoordinates of –1 and 1.
Graphing y=Acot(Bx-C)
1. Find two consecutive asymptotes by setting the
variable expression in the cotangent equal to 0
and ˝ and solving
Bx – C = 0 and Bx – C = 
Bx – C = 
2. Identify an x-intercept, midway between
consecutive asymptotes.
x
3. Find the points on the graph 1/4 and 3/4 of the
y-coordway between an x-intercept and the asymptotes.
inate is -A.
These points have y-coordinates of –A and A.
4. Use steps 1-3 to graph one full period of the
function. Add additional cycles to the left or right
as needed.
y = A cot (Bx – C)
y-coordinate is A.
x-intercept
between
asymptotes
Bx – C
=0
Example
Graph y = 2 cot 3x
Solution:
3x=0 and 3x=
x=0 and x = /3 are vertical asymptotes
An x-intercepts occurs between 0 and /3 so an xintercepts is at (/6,0)
The point on the graph midway between the
asymptotes and intercept are /12 and 3/12.
These points have y-coordinates of -A and A or -2
and 2
Graph one period and extend as needed
Example cont
• Graph y = 2 cot 3x
10
8
6
4
2
-3
-2
-1
1
-2
-4
-6
-8
-10
2
3
The Cosecant Curve: The Graph
of y = cscx and Its Characteristics
y
-  /2 1
-2
-3  /2
-˝
-1
3  /2
 /2
˝
2
x
Characteristics
Period: 2
Domain: All real numbers except
integral multiples of 
Range: All real numbers y such that
y < -1 or y > 1
Vertical asymptotes: at integral
multiples of 
Odd function with origin symmetry
The Secant Curve: The Graph of
y=secx and Its Characteristics
y
1
-2
-3  /2
-
-˝ /2
-1
˝  /2

3  /2
2
x
Characteristics
Period: 2
Domain: All real numbers except odd
multiples of  /2
Range: All real numbers y such that
y < -1 or y > 1
Vertical asymptotes: at odd multiples
of / 2
Even function with origin symmetry
Text Example
Use the graph of y = 2 sin 2x to obtain the graph of y = 2 csc 2x.
y
y
2
2
˝
-˝
-2
x
˝
x
-2
Solution The x-intercepts of y = 2 sin 2x correspond to the vertical
asymptotes of y = 2 csc 2x. Thus, we draw vertical asymptotes through the xintercepts. Using the asymptotes as guides, we sketch the graph of y = 2 csc 2x.
Graphs of Other
Trigonometric
Functions