4.5 – graphs of the other trigonometric functions tangent and cotangent
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4.5 – Graphs of the other Trigonometric
Functions Tangent and Cotangent
In this section, you will learn to:•Sketch the graphs of tangent and cotangent functions
The graph of the tangent curve:
•The graph of tan :y x
90 180 270 360-90-180-270-360
1
2
3
4
5
-1
-2
-3
-4
-5
x
y
The graph of the tangent curve:• The graph of tan :y x
Period :
Domain : all real numbers
except2
x n
Range : ,
Asymptotes :2
x n
The graph of the cotangent curve:
•The graph of cot :y x
90 180 270 360-90-180-270-360
1
2
3
4
5
-1
-2
-3
-4
-5
x
y
The graph of the cotangent curve:
Period :
Domain : all real numbers
except x n
Range : ,
Asymptotes : x n
• The graph of cot :y x
Question: Is there an amplitude for a tangent or a cotangent function? Why or why not?
No amplitude, since the two curves extend infinitely in both directions.
Graphical effects of constants , , and in
tan and cot
functions :
a b c d
y a bx c d y a bx c d
Period of Tangent and Cotangent Functions:
The period of tangent and cotangent functions
tan and cot
is .
y a bx c d y a bx c d
b
1 a) Period of 2 tan 1:
2 4y x
21
2b
THIS IS DIFFERENT
Where do you think you need to set the left and right endpoints for a tangent graph below?
Asymptotes of the tangent graph function:
Where do you think you need to set the left and right endpoints for a cotangent graph below?
Asymptotes of the cotangent graph function:
•If a is positive, then there is no reflection about the x-axis.
•If a is negative, then there is a reflection about the x-axis.
Reflection:
Reflection : tany a bx c d
45 90-45-90
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-2
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-4
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x
y
tany x tany x
45 90-45-90
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-1
-2
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-4
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x
y
Reflection : tany a bx c d
andan nt tay xy x
45 90-45-90
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2
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4
5
-1
-2
-3
-4
-5
x
y
Reflection : coty a bx c d
coty x coty x
45 90 135 180
1
2
3
4
5
-1
-2
-3
-4
-5
x
y
45 90 135 180
1
2
3
4
5
-1
-2
-3
-4
-5
x
y
Reflection : coty a bx c d
andot tc coy xy x
45 90 135 180
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2
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4
5
-1
-2
-3
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x
y
Effects of a on the tangent and cotangent graphs:
2tany x1
tan2
y x
45 90-45-90
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4
5
-1
-2
-3
-4
-5
x
y
45 90-45-90
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-1
-2
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-4
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x
y
a) If a > 1, then the graph rises faster.
b) If 0< a < 1, then the graph rises slower.
Effects of a on the tangent and cotangent graphs:
Effects of c on the tangent and cotangent graphs:
tan4
y x
tan
4y x
45 90-45-90-135
1
2
3
4
5
-1
-2
-3
-4
-5
x
y
45 90 135-45-90
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2
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5
-1
-2
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-4
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x
y
The constant c determines the phase shift of the graph.
Phase Shift = - c/k (or –c/b)
a) If c is positive, then the shift is toward the left.
b) If c is negative, then the shift is toward the right.
Horizontal Translation or Phase Shift:
Effects of d on the tangent and cotangent graphs:
tan 2y x tan 2y x
45 90-45-90
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2
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5
-1
-2
-3
-4
-5
x
y
45 90-45-90
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-1
-2
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-4
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x
y
The constant d determines the verticaltranslation of the graph.
a) If d is positive, then the vertical shift is upward.
b) If d is negative, then the vertical shift is downward.
Vertical Translation:
Find the amplitude, period, reflections, horizontal shift, vertical shift, endpoints and sketch the graph.
EXAMPLE : 2cot 2 34
y x
a) Amplitude:
b) Period:
c) Horizontal Translation:
d) Vertical Shift:
e) Reflection:
2b
to the right8
3 units downward
about the -axisx
EXAMPLE : 2cot 2 34
y x
none
Problem : 2cot 2 34
y x
e) Endpoints:
Verify distance with the period:
2 0 24 4
x and x
5
8 8 2
52 2
4 45
8 8
x x
x x
Graph of 2cot 2 34
y x
The graph of the secant curve:• The graph of sec :y x
45 90 135 180 225 270 315 360
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2
3
4
5
-1
-2
-3
-4
-5
x
y
The graph of the secant curve:
The graph of sec :y x Period : 2
Domain : all real numbers
except2
x n
Range : , 1 1,and
Asymptotes :2
x n
The graph of the cosecant curve:
• The graph of csc :y x
45 90 135 180 225 270 315 360
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2
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5
-1
-2
-3
-4
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x
y
The graph of the cosecant curve:The graph of csc :y x
Period : 2
Domain : all real numbers
except x n
Range : , 1 1,and
Asymptotes : x n
a) Is there an amplitude for a secant or a cosecant function? Why or why not?
Graphical effects of constants , , and in
sec and csc
functions :
a b c d
y a bx c d y a bx c d
b) Period is
c) The horizontal translation, vertical translation, and reflection all stay the same.
d) It is best to sketch the cosecant and secant graph by first graphing the reciprocal functions of sine and cosine.
2.
b
Graphical effects of constants , , and in
sec and csc
functions :
a b c d
y a bx c d y a bx c d
Find the amplitude, period, reflections, horizontal shift, vertical shift, endpoints and sketch the graph.
EXAMPLE : 2csc 2 34
y x
a) Amplitude:
b) Period:
c) Horizontal Translation:
d) Vertical Shift:
e) Reflection:
2 2
2b
to the right8
3 units downward
about the -axisx
none
EXAMPLE : 2csc 2 34
y x
Problem : 2csc 2 34
y x
e) Endpoints:
Verify distance with the period:
2 0 2 24 4
x and x
9
8 8
92 2
4 49
8 8
x x
x x
Graph of 2csc 2 34
y x
-axis Reflection : siny y a bx c d
siny x siny x
90 180 270 360-90-180-270-360
1
2
-1
-2
x
y
90 180 270 360-90-180-270-360
1
2
-1
-2
x
y
-axis Reflection : siny y a bx c d
andin ns siyy x x
90 180 270 360-90-180-270-360
1
2
-1
-2
x
y
-axis Reflection : cosy y a bx c d
cosy x cosy x
90 180 270 360-90-180-270-360
1
2
-1
-2
x
y
90 180 270 360-90-180-270-360
1
2
-1
-2
x
y
-axis Reflection : cosy y a bx c d
andos sc coyy x x
90 180 270 360-90-180-270-360
1
2
-1
-2
x
y
-axis Reflection : tany y a bx c d
tany x tany x
45 90-45-90
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-2
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x
y
45 90-45-90
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x
y
andan nt tay xy x
45 90-45-90
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-2
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x
y
-axis Reflection : tany y a bx c d
coty x
90 180
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x
y
-90-180
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x
y
coty x
-axis Reflection : coty y a bx c d
-axis Reflection : coty y a bx c d
andot tc coyy x x
90 180-90-180
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-1
-2
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-4
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x
y
secy x secy x
90 180 270 360
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-1
-2
-3
-4
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x
y
-90-180-270-360
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x
y
-axis Reflection : secy y a bx c d
-axis Reflection : secy y a bx c d
andec cs seyy x x
90 180 270 360-90-180-270-360
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x
y
cscy x cscy x -axis Reflection : cscy y a bx c d
90 180 270 360
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-1
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x
y
-90-180-270-360
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-1
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-4
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x
y
-axis Reflection : cscy y a bx c d
andsc cc csyy x x
90 180 270 360-90-180-270-360
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2
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-1
-2
-3
-4
-5
x
y
• If a is positive, then there is no reflection about the x-axis.
• If a is negative, then there is a reflection about the x-axis.
• If b is positive, then there is no reflection about the y-axis.
• If b is negative, then there is a reflection about the y-axis.
Reflection:
Example : 2sin 2 1y x
-90-180
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x
y
Example : 2sin 2 1y x
-90-180
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x
y
2sin 2 1y x
-90-180
1
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x
y
2sin 2 1y x
-90-180
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x
y
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