Graphs of Other
Trigonometric
Functions
The Tangent Curve: The Graph
of y=tanx and Its Characteristics
y
Period: π
Domain: All real numbers
except π /2 + k π,
k an integer
1
–2 π
5π
–
2
–π
π
3π
–
2
0
–π
2
–1
Range: All real numbers
2π
π
π
2
3π
2
5π
2
x
Symmetric with respect to
the origin
Vertical asymptotes at
odd multiples of π /2
Graphing y = A tan(Bx – C)
y = A tan (Bx – C)
Bx – C = - π/2
x-intercept
between
asymptotes
1.
1. Find
Find two
two consecutive
consecutive asymptotes
asymptotes by
by setting
setting the
the
variable
variable expression
expression in
in the
the tangent
tangent equal
equal to
to -π/2
-π/2
and
and π/2
π/2 and
and solving
solving
Bx – C = π /2
Bx
Bx –– C
C == -π/2
-π/2 and
and Bx
Bx –– C
C == π/2
π/2
2.
2. Identify
Identify an
an x-intercept,
x-intercept, midway
midway between
between
consecutive
consecutive asymptotes.
asymptotes.
x
3.
3. Find
Find the
the points
points on
on the
the graph
graph 1/4
1/4 and
and 3/4
3/4 of
of the
the
way
way between
between and
and x-intercept
x-intercept and
and the
the asymptotes.
asymptotes.
These
These points
points have
have y-coordinates
y-coordinates of
of –A
–A and
and A.
A.
4.
4. Use
Use steps
steps 1-3
1-3 to
to graph
graph one
one full
full period
period of
of the
the
function.
function. Add
Add additional
additional cycles
cycles to
to the
the left
left or
or right
right
as
as needed.
needed.
1
Text Example
Graph y = 2 tan x/2 for – π< x < 3 π
Solution
Step 1 Find two consecutive asymptotes.
Thus, two consecutive asymptotes occur at x = - π and x = π.
Step 2 Identify any x-intercepts, midway between consecutive
asymptotes. Midway between x = -π and x = π 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 = -π and
x = π, 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 –π to π. In order to
graph for –π < x < 3 π, we continue the
pattern and extend the graph another
full period on the right.
y = 2 tan x/2
y
4
2
-˝
˝
3˝
x
-2
-4
The Cotangent Curve: The Graph
of y = cotx and Its Characteristics
The
The Graph
Graph of
of yy == cot
cot xx and
and Its
Its Characteristics
Characteristics
y
4
2
-π
-
π /2
π /2
-2
-4
˝
3
π /2
2π
x
Characteristics
Characteristics
Period:
Period: π
π
Domain:
Domain: All
All real
real numbers
numbers except
except
integral
integral multiples
multiples of
of π
π
Range:
Range: All
All real
real numbers
numbers
Vertical
Vertical asymptotes:
asymptotes: at
at integral
integral
multiples
π
multiples of
of π
Α
Αnn x-intercept
x-intercept occurs
occurs midway
midway between
between
each
each pair
pair of
of consecutive
consecutive asymptotes.
asymptotes.
Odd
Odd function
function with
with origin
origin symmetry
symmetry
Points
Points on
on the
the graph
graph 1/4
1/4 and
and 3/4
3/4 of
of the
the way
way
between
between consecutive
consecutive asymptotes
asymptotes have
have yycoordinates
coordinates of
of –1
–1 and
and 1.
1.
2
Graphing y=Acot(Bx-C)
1.
1. Find
Find two
two consecutive
consecutive asymptotes
asymptotes by
by setting
setting the
the
variable
variable expression
expression in
in the
the cotangent
cotangent equal
equal to
to 00
and
and ˝˝ and
and solving
solving
Bx
Bx – C = π
Bx –– C
C == 00 and
and Bx
Bx –– C
C == ππ
2.
2. Identify
Identify an
an x-intercept,
x-intercept, midway
midway between
between
consecutive
asymptotes.
consecutive asymptotes.
x
3.
Find
the
points
on
the
graph
1/4
3. Find the points on the graph 1/4 and
and 3/4
3/4 of
of the
the
y-coordway between an x-intercept and the asymptotes.
inate is -A. way between an x-intercept and the asymptotes.
These
points
have
y-coordinates
of
–A
and
A.
These points have y-coordinates of –A and A.
4.
4. Use
Use steps
steps 1-3
1-3 to
to graph
graph one
one full
full period
period of
of the
the
function.
function. Add
Add additional
additional cycles
cycles to
to the
the left
left or
or right
right
as
as needed.
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=π
x=0 and x = π/3 are vertical asymptotes
An x-intercepts occurs between 0 and π/3 so an xintercepts is at (π/6,0)
The point on the graph midway between the
asymptotes and intercept are π/12 and 3π/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
3
-2
-4
-6
-8
-10
3
The Cosecant Curve: The Graph
of y = cscx and Its Characteristics
y
- π /2 1
-2π
-3 π /2
-˝
3 π /2
π /2
-1
˝
2π
x
Characteristics
Characteristics
Period:
Period: 2π
2π
Domain:
Domain: All
All real
real numbers
numbers except
except
integral
integral multiples
multiples of
of ππ
Range:
Range: All
All real
real numbers
numbers yy such
such that
that
yy <
< -1
-1 or
or yy >
> 11
Vertical
Vertical asymptotes:
asymptotes: at
at integral
integral
multiples
multiples of
of ππ
Odd
Odd function
function with
with origin
origin symmetry
symmetry
The Secant Curve: The Graph of
y=secx and Its Characteristics
y
1
-2π
-3 π /2
-˝π /2
-π
-1
˝π / 2
π
3 π /2
2π
x
Characteristics
Characteristics
Period:
Period: 2π
2π
Domain:
Domain: All
All real
real numbers
numbers except
except odd
odd
multiples
multiples of
of ππ //22
Range:
Range: All
All real
real numbers
numbers yy such
such that
that
yy <
-1 or
or yy >
> 11
< -1
Vertical
Vertical asymptotes:
asymptotes: at
at odd
odd multiples
multiples
of
of π/
π/ 22
Even
Even function
function with
with origin
origin symmetry
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.
4
Graphs of Other
Trigonometric
Functions
5