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MATHPOWERTM 12, WESTERN EDITION
4.6
4.6.1
Chapter 4 Trigonometric Functions
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The graphs of y = tanand the reciprocal trigonometricfunctions are periodic functions, but they differ from thesinusoidal functions y = sin and y = cos in that they haverestrictions on their domains.
The Reciprocal Functions
Recall the reciprocal functions:
csc x
1
sin xsec x
1
cos x tan x
1
cot x
4.6.2
The transformations that will be applied to the reciprocal functions follow the same rules as those of the primary ratios.
y = acsc b(x + c) + d
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4.6.3
Graphing the Tangent Function
Asymptotes:
2
,32
,52
,...,2
n, n I
Domain: {x | x
2
n, n I , x R}
Range:
Period:
all real numbers
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1. Draw verticalasymptotes.
2. Plot thepoints aty = 1 and-1.3. Sketchthe cotangentfunction.
Graphing the Cotangent Function
4.6.4
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4.6.5
Graphing the Cotangent Function [cont’d]
Asymptotes: , 2,3, ..., n, n I
Domain:
Range:
Period:
{x | x n, n I , x R}
all real numbers
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4.6.6
Graphing the Cosecant Function
1. Draw theverticalasymptotes.
2. Plot thepoints at y = 1 and-1.3. Sketchthe cosecantfunction.
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Graphing the Cosecant Function [cont’d]
Asymptotes: , 2,3, ..., n, n I
Domain:
Range:
Period:
-1 ≥ y ≥ 1
4.6.7
{x | x n, n I , x R}
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4.6.8
The Cosecant FunctionGraph y = 2csc x.
y = csc x
y = 2csc x
Asymptotes: , 2,3, ..., n, n I
Domain:
Range:
Period:
-2 ≥ y ≥ 2
For y = 2csc x:
Graph y = csc 2x.
For y = csc2 x:
Asymptotes:
2
, ,32
2, ...,2
2
n, n I
Domain:
Range: -1 ≥ y ≥ 1
Period:
{x | x n, n I , x R} {x | x
2
2
n, n I , x R}
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4.6.9
Graphing the Secant Function
1. Draw theverticalasymptotes.
2. Plot thepoints at y = 1 and-1.3. Sketchthe secantfunction.
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Graphing the Secant Function [cont’d]
Asymptotes:
Domain: {x | x
2
n, n I , x R}
Range:
Period:
-1 ≥ y ≥ 1
4.6.10
2
,32
,52
,...,2
n, n I
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4.6.11
Suggested Questions:Page 232 1-11 odd,12-15 (graphing calculator),16, 17