MATHPOWERTM 12, WESTERN EDITION

4.6

4.6.1

Chapter 4 Trigonometric Functions

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

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

1. Draw verticalasymptotes.

2. Plot thepoints aty = 1 and-1.3. Sketchthe cotangentfunction.

Graphing the Cotangent Function

4.6.4

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

4.6.6

Graphing the Cosecant Function

1. Draw theverticalasymptotes.

2. Plot thepoints at y = 1 and-1.3. Sketchthe cosecantfunction.

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}

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}

4.6.9

Graphing the Secant Function

1. Draw theverticalasymptotes.

2. Plot thepoints at y = 1 and-1.3. Sketchthe secantfunction.

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

4.6.11

Suggested Questions:Page 232 1-11 odd,12-15 (graphing calculator),16, 17