reciprocal trig fns
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Graphing the reciprocal trig functions.
csc , sec , coty x y x y x
Example 1. Graph csc , first graph sin , then use
reciprocals to find the graph of csc .
y x y x
y x
Here is the graph of siny x
reciprocalWe want the trig function
cscy x
1So we take , the reciprocal of all
sin
the trig values of sin
yx
y x
Sine function: 1 1, the only problem
with reciprocals here is when ? ? ? ? ? ?
y
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1The values are equal to 0, is undefined and here we have asymptotes!
0y
Vertical Asymptotes!
At: 2 , ,0, , 2 etc.x
1 1
1 1 0.1 10
1 2 0.05 20
2
y yy y
1 1
1 1 0.1 10
1 2 0.05 20
2
y yy y
This gives us the graph:
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We can do the same thing for the graph of: sec ,
that is graph cos first, then draw the vertical
asymptotes where cos 0 and sketch sec .
y x
y x
x y x
Graph cos to get:y x
Now for the asymptotes:Now use camel technique:
draw on the humps!
Now for coty x
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Graph tan first, then draw the vertical
asymptotes where tan 0 and sketch cot .
y x
x y x
Now the vertical asymptotes appear
where the tangent function 0
Now where the tangent function
had asymptotes, the cot function 0
Now where the tan 1 or tan 1,
cot 1 or cot 1
x x
x x
Now join the points between the
asymptotes!
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Properties of the reciprocal trig functions:
I. csc , Domain: , , , Integers
Range: 1 1, asymptotes: ,
y x x n n
y y x n n I
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II. sec , Domain: , , , Integers2
Range: 1 1, asymptotes: , , odd2
ny x x n
ny y x n I
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III. cot , Domain: , , , Integers
Range: , , asymptotes: , ,
y x x n n
x n n I
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Suppose we were asked to graph: 2sec 2 1.y x
The center of the graph is: 1, where the graph crosses this
line is where the asymptotes occur!
y
2We would graph: 2cos 2 1, , ,
2 42, no phase shift, down 1
y x P I
A
The starting point would be: 0, 0,1 this is high point, then
right down 2, right down 2,right up 2,right up 2,4 4 4 4etc
a d
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Now we can
join the points.
The starting point would be: 0, 0,1 this is high point, then
right down 2, right down 2,right up 2,right up 2,4 4 4 4etc
a d
Center is 1y
Now where the
graph crosses the
center line is
where our vertical
asymptotes occur!
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Now we can use the camel technique to sketch the graph on the humps
of the cosine graph!