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Pre-CalculusChapter 4

Trigonometric Functions

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4.6 Graphs of Other Trigonometric Functions

Objectives: Sketch the graphs of tangent functions. Sketch the graphs of cotangent functions.

Sketch the graphs of secant and cosecant functions.

Sketch the graphs of damped trigonometric functions.

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Tangent and Cotangent Complete the worksheet

“Graphs of Tangent and Cotangent”

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Graph of Tangent Domain:

Range:

Period:

Zeros:

Vertical Asymptotes:

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Graph of Cotangent Domain:

Range:

Period:

Zeros:

Vertical Asymptotes:

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General Form of Tangent and Cotangent

The general form of tangent and cotangent are: y = a tan (bx – c)

y = a cot (bx – c)

Where π/b = period c = horizontal shift

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Graph Transformation of Tangent To sketch the graph of y = a tan (bx – c):

1. Sketch two asymptotes.bx – c = –π/2 and bx – c = π/2

2. Plot the zeros.Midpoint between the

asymptotes.3. Plot additional points.

Midway between zero & VA = ± a.

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Graph Transformation of Cotangent To sketch the graph of y = a cot (bx – c):

1. Sketch two asymptotes.bx – c = 0 and bx – c = π

2. Plot the zeros.Midpoint between the

asymptotes.3. Plot additional points.

Midway between zero & VA = ± a.

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Examples

Sketch the graph of each function by

hand.

1. y = tan x/2

2. y = –3 tan 2x

3. y = 2 cot x/3

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Secant and Cosecant

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Secant and Cosecant State the reciprocal function for

each

sec x =

csc x =

We will use the reciprocal functions to graph secant and cosecant.

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Cosecant Where do the zeros of sin x occur?

If sin x is zero at these values, then what happens to csc x?

Where do the maximum values of sin x occur? What is the max value? What about minimum values?

What does this mean for csc x?

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Graph Cosecant

To graph csc x, Graph sin x. Sketch VA. Find max/min.

Create curve.

                         

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Secant Where do the zeros of cos x occur?

If cos x is zero at these values, then what happens to sec x?

Where do the maximum values of cos x occur? What is the max value? What about minimum values?

What does this mean for sec x?

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Graph Secant

To graph sec x, Graph cos x. Sketch VA. Find max/min.

Create curve.

                         

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Properties of sec x and csc x

sec x csc x

Period

Domain

Range

VA

Symmet

ry

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Example 2 Sketch the graph of

by hand. (Hint: graph sine first.)

4csc2

xy

                                         

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Example 3 Sketch the graph of

by hand. (Hint: graph cosine first.) 24sec2 xy

                                         

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Homework 4.6 Worksheet 4.6