Graphs of Other Trigonometric Functions

Copyright © 2012 Pearson Education, Inc. Publishing as

Prentice Hall.

6

We are interested in the graph of y = tan x

Start with a "t" chart and let's choose values from our unit circle and find the tangent values. Tangent has a period of so it will repeat every .

x y = tan x

3

2

undefined

73.13

4

1

6

58.0

3

3

x

y

2

3

2

3

6

would mean there is a vertical asymptote here

6

y = tan xLet's choose more values.

x y = tan x

6

0 0

73.13

4

1

3

58.03

3

x

y

2

3

2

3

6

would mean there is a vertical asymptote here2

undefined

Since we went from we have one complete period 2

to2

y

x

2

3

2

32

2

period:

The Unit Circle and Tangent Applet

The red vertical lines are not part of the graph but are the asymptotes.

Let's see what the graph would look like for y = tan x for 3 complete periods.

Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.

Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.

Steps in Graphing

1. Find two consecutive vertical asymptotes by finding an interval containing one period. Solve for x.

2. Sketch the two vertical asymptotes found in Step 1.

4. Find the points on the graph and of the way between the consecutive asymptotes. These points have y-coordinates of and , respectively.

5. Join the points with a smooth curve, approaching the vertical asymptotes. Indicate additional asymptotes and periods of the graph to the left or right as needed.

3. Identify an x-intercept, midway between the consecutive asymptotes.

Graph for

Step 1: Find two consecutive asymptotes.

Since , we will set up the following:

An interval containing one period is . Two consecutive asymptotes occur at and

Step 2: Identify an x-intercept, midway between the consecutive asymptotes.

Midway between and is . An x-intercept is 0, and the graph passes through .

Step 3: Find points on the graph and of the way between the consecutive asymptotes. These points have y-coordinates of and .

The coefficient, , of the function is 2. Therefore the y-coordinates are and . The graph passes through and .

Step 4: Use steps 1 – 3 to graph one full period of the function. In order to graph for , we continue the pattern and extend the graph another full period to the right.

See figure 4.79 on page 539.

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 12

1. Find consecutive vertical asymptotes by solving for x:

5. Sketch one branch and repeat.

Find the period and asymptotes and sketch the graph

of xy 2tan3

1

2. Vertical asymptotes are and

x

xy 2tan3

1

8

3

1 0

08

3

18

3

3

1

y

x2

8

3

4

x

4

x

3

1,

8

3

1,

8

3

1,

8

3

4. Find points on the graph and of the way between the consecutive asymptotes.

3. Identify an x-intercept, midway between the consecutive asymptotes: x-intercept =

Graph xy4

1tan2

Vertical asymptotes are

24

1

24

1 xandx

22 xandx

Divide the interval from - 2 to 2 into four equal parts and plot points.

y

x

x = - 2 x = 2

x

2 0

0

2 2

3

xy4

1tan2

The midway x-intercept is

Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.

y = cot x

Again the vertical lines are not part of the graph but are the asymptotes.

Let's look at the tangent graph again to compare these.

Notice vertical asymptotes of one are zeros of the other.

y = tan x

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 16

Graph of the Cotangent Function

2. range: (–, +) 3. period: 4. vertical asymptotes:

kkx

1. domain : all real #’s kkx

Properties of y = cot x

y

x

2

2

2

32

3

2

xy cot

0xvertical asymptotes xx 2x

To graph y = cot x, use the identity .x

xx

sin

coscot

At values of x for which sin x = 0, the cotangent function is undefined and its graph has vertical asymptotes.

Steps in Graphing

1. Find two consecutive vertical asymptotes by finding an interval containing one period. Solve for x.

2. Sketch the two vertical asymptotes found in Step 1.

4. Find the points on the graph and of the way between the consecutive asymptotes. These points have y-coordinates of and , respectively.

5. Join the points with a smooth curve, approaching the vertical asymptotes. Indicate additional asymptotes and periods of the graph to the left or right as needed.

3. Identify an x-intercept, midway between the consecutive asymptotes.

0<𝐵𝑥−𝐶<𝜋

Graph

Step 1: Find two consecutive asymptotes.

An interval containing one period is . The two consecutive asymptotes are and .

Step 2: Identify an x-intercept midway between the consecutive asymptotes.

The x-intercept midway between is . The x-intercept is .

Step 3: Find points on the graph and of the way between the consecutive asymptotes. These points have y-coordinates of and .

The coefficient, , of the function is 3. Therefore the y-coordinates are and 3. The graph passes through and (

Step 4: Use steps 1 – 3 to graph one full period of the function.

See figure 4.81 on page 542.

Cosecant is the reciprocal of sine

One period: 2π

π 2π 3π0

−π−2π−3π

Vertical asymptotes where sin θ = 0

θ

csc θ

sin θ

2

3

x

2

2

2

2

5

y

4

4

Graph of the Cosecant Function

2. range: (–,–1] [1, +)

3. period:

where sine is zero.

4. vertical asymptotes: kkx

1. domain : all real x kkx

sin

1csc

xx To graph y = csc x, use the identity .

Properties of y = csc x xy csc

xy sin

At values of x for which sin x = 0, the cosecant function

is undefined and its graph has vertical asymptotes.

For the graph of y = f(x) = csc x, we'll take the reciprocals of the sine values.

x sin x y = cosec x

6

0 0

2

1

2

1

6

5

When we graph these rather than plot points after we see this, we'll use the sine graph as a sketching aid and then get the cosecant graph.

x

y

1

- 1

undefined

2

1

2

12

y = f(x) = csc xchoose more values

x sin x y = cosec x

6

7 0

2

1

2

31

6

112

1

We'll use the sine graph as the sketching aid.

x

y

1

- 12 0

6

2

undefined

2

1

2

undefined

When the sine is 0 the cosecant will have an asymptote.

Using a Sine Curve to Obtain a Cosecant Curve

Graph

Step 1: Graph

Use dash lines (color is preferred) to represent the sine graph.

Step 2: Use the x-intercepts to represent the vertical asymptotes of the cosecant graph.

Step 3: Sketch the graph as shown in figure 4.84 on page 543.

Secant is the reciprocal of cosine

One period: 2π

π 3π−2π 2π−π−3π 0θ

sec θ

cos θ

Vertical asymptotes where cos θ = 0

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2

3

y

x

2

2

2 3

2

5

4

4

xy cos

Graph of the Secant Function

2. range: (–,–1] [1, +) 3. period: 4. vertical asymptotes:

kkx 2

1. domain : all real x)(

2 kkx

cos

1sec

xx The graph y = sec x, use the identity .

Properties of y = sec x

xy sec

At values of x for which cos x = 0, the secant function is undefined and its graph has vertical asymptotes.

Graph for

Step 1: Graph

This equation is in the form .

The amplitude,

The period

Divide into 4 quarter-periods for the five key points. Starting with , the five x-values are .

So the coordinates for the key points are .

Use these key points to graph the cosine equation, using dashed lines.

Step 2: Use the x-intercepts to represent the vertical asymptotes of the secant graph.

Step 3: Sketch the graph as shown in figure 4.85 on page 544.