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4.5 – Graphs of the other Trigonometric Functions Tangent and Cotangent

# 4.5 – Graphs of the other Trigonometric Functions Tangent and Cotangent

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4.5 – Graphs of the other Trigonometric

Functions Tangent and Cotangent

In this section, you will learn to:•Sketch the graphs of tangent and cotangent functions

The graph of the tangent curve:

•The graph of tan :y x

90 180 270 360-90-180-270-360

1

2

3

4

5

-1

-2

-3

-4

-5

x

y

The graph of the tangent curve:• The graph of tan :y x

Period :

Domain : all real numbers

except2

x n

Range : ,

Asymptotes :2

x n

The graph of the cotangent curve:

•The graph of cot :y x

90 180 270 360-90-180-270-360

1

2

3

4

5

-1

-2

-3

-4

-5

x

y

The graph of the cotangent curve:

Period :

Domain :  all real numbers

except x n

Range : ,

Asymptotes : x n

• The graph of cot :y x

Question: Is there an amplitude for a tangent or a cotangent function? Why or why not?

No amplitude, since the two curves extend infinitely in both directions.

Graphical effects of constants   , , and   in

tan  and cot

functions :

a b c d

y a bx c d y a bx c d

Period of Tangent and Cotangent Functions:

The period of tangent and cotangent functions

tan   and   cot

is .

y a bx c d y a bx c d

b

1  a) Period of 2 tan 1:

2 4y x

21

2b

THIS IS DIFFERENT

Where do you think you need to set the left and right endpoints for a tangent graph below?

Asymptotes of the tangent graph function:

Where do you think you need to set the left and right endpoints for a cotangent graph below?

Asymptotes of the cotangent graph function:

•If a is positive, then there is no reflection about the x-axis.

•If a is negative, then there is a reflection about the x-axis.

Reflection:

Reflection : tany a bx c d

45 90-45-90

1

2

3

4

5

-1

-2

-3

-4

-5

x

y

tany x tany x

45 90-45-90

1

2

3

4

5

-1

-2

-3

-4

-5

x

y

Reflection : tany a bx c d

andan nt tay xy x

45 90-45-90

1

2

3

4

5

-1

-2

-3

-4

-5

x

y

Reflection : coty a bx c d

coty x coty x

45 90 135 180

1

2

3

4

5

-1

-2

-3

-4

-5

x

y

45 90 135 180

1

2

3

4

5

-1

-2

-3

-4

-5

x

y

Reflection : coty a bx c d

andot tc coy xy x

45 90 135 180

1

2

3

4

5

-1

-2

-3

-4

-5

x

y

Effects of a on the tangent and cotangent graphs:

2tany x1

tan2

y x

45 90-45-90

1

2

3

4

5

-1

-2

-3

-4

-5

x

y

45 90-45-90

1

2

3

4

5

-1

-2

-3

-4

-5

x

y

a) If a > 1, then the graph rises faster.

b) If 0< a < 1, then the graph rises slower.

Effects of a on the tangent and cotangent graphs:

Effects of c on the tangent and cotangent graphs:

tan4

y x

tan

4y x

45 90-45-90-135

1

2

3

4

5

-1

-2

-3

-4

-5

x

y

45 90 135-45-90

1

2

3

4

5

-1

-2

-3

-4

-5

x

y

The constant c determines the phase shift of the graph.

Phase Shift = - c/k (or –c/b)

a) If c is positive, then the shift is toward the left.

b) If c is negative, then the shift is toward the right.

Horizontal Translation or Phase Shift:

Effects of d on the tangent and cotangent graphs:

tan 2y x tan 2y x

45 90-45-90

1

2

3

4

5

-1

-2

-3

-4

-5

x

y

45 90-45-90

1

2

3

4

5

-1

-2

-3

-4

-5

x

y

The constant d determines the verticaltranslation of the graph.

a) If d is positive, then the vertical shift is upward.

b) If d is negative, then the vertical shift is downward.

Vertical Translation:

Find the amplitude, period, reflections, horizontal shift, vertical shift, endpoints and sketch the graph.

EXAMPLE : 2cot 2 34

y x

a) Amplitude:

b) Period:

c) Horizontal Translation:

d) Vertical Shift:

e) Reflection:

2b

to the right8

3 units downward

EXAMPLE : 2cot 2 34

y x

none

Problem : 2cot 2 34

y x

e) Endpoints:

Verify distance with the period:

2 0 24 4

x and x

5

8 8 2

52 2

4 45

8 8

x x

x x

Graph of 2cot 2 34

y x

The graph of the secant curve:• The graph of sec :y x

45 90 135 180 225 270 315 360

1

2

3

4

5

-1

-2

-3

-4

-5

x

y

The graph of the secant curve:

The graph of sec :y x Period : 2

Domain : all real numbers

except2

x n

Range : , 1 1,and

Asymptotes :2

x n

The graph of the cosecant curve:

• The graph of csc :y x

45 90 135 180 225 270 315 360

1

2

3

4

5

-1

-2

-3

-4

-5

x

y

The graph of the cosecant curve:The graph of csc :y x

Period : 2

Domain : all real numbers

except x n

Range : , 1 1,and

Asymptotes : x n

a) Is there an amplitude for a secant or a cosecant function? Why or why not?

Graphical effects of constants   , , and   in

sec  and csc

functions :

a b c d

y a bx c d y a bx c d

b) Period is

c) The horizontal translation, vertical translation, and reflection all stay the same.

d) It is best to sketch the cosecant and secant graph by first graphing the reciprocal functions of sine and cosine.

2.

b

Graphical effects of constants   , , and   in

sec  and csc

functions :

a b c d

y a bx c d y a bx c d

Find the amplitude, period, reflections, horizontal shift, vertical shift, endpoints and sketch the graph.

EXAMPLE : 2csc 2 34

y x

a) Amplitude:

b) Period:

c) Horizontal Translation:

d) Vertical Shift:

e) Reflection:

2 2

2b

to the right8

3 units downward

none

EXAMPLE : 2csc 2 34

y x

Problem : 2csc 2 34

y x

e) Endpoints:

Verify distance with the period:

2 0 2 24 4

x and x

9

8 8

92 2

4 49

8 8

x x

x x

Graph of 2csc 2 34

y x

-axis Reflection : siny y a bx c d

siny x siny x

90 180 270 360-90-180-270-360

1

2

-1

-2

x

y

90 180 270 360-90-180-270-360

1

2

-1

-2

x

y

-axis Reflection : siny y a bx c d

andin ns siyy x x

90 180 270 360-90-180-270-360

1

2

-1

-2

x

y

-axis Reflection : cosy y a bx c d

cosy x cosy x

90 180 270 360-90-180-270-360

1

2

-1

-2

x

y

90 180 270 360-90-180-270-360

1

2

-1

-2

x

y

-axis Reflection : cosy y a bx c d

andos sc coyy x x

90 180 270 360-90-180-270-360

1

2

-1

-2

x

y

-axis Reflection : tany y a bx c d

tany x tany x

45 90-45-90

1

2

3

4

5

-1

-2

-3

-4

-5

x

y

45 90-45-90

1

2

3

4

5

-1

-2

-3

-4

-5

x

y

andan nt tay xy x

45 90-45-90

1

2

3

4

5

-1

-2

-3

-4

-5

x

y

-axis Reflection : tany y a bx c d

coty x

90 180

1

2

3

4

5

-1

-2

-3

-4

-5

x

y

-90-180

1

2

3

4

5

-1

-2

-3

-4

-5

x

y

coty x

-axis Reflection : coty y a bx c d

-axis Reflection : coty y a bx c d

andot tc coyy x x

90 180-90-180

1

2

3

4

5

-1

-2

-3

-4

-5

x

y

secy x secy x

90 180 270 360

1

2

3

4

5

-1

-2

-3

-4

-5

x

y

-90-180-270-360

1

2

3

4

5

-1

-2

-3

-4

-5

x

y

-axis Reflection : secy y a bx c d

-axis Reflection : secy y a bx c d

andec cs seyy x x

90 180 270 360-90-180-270-360

1

2

3

4

5

-1

-2

-3

-4

-5

x

y

cscy x cscy x -axis Reflection : cscy y a bx c d

90 180 270 360

1

2

3

4

5

-1

-2

-3

-4

-5

x

y

-90-180-270-360

1

2

3

4

5

-1

-2

-3

-4

-5

x

y

-axis Reflection : cscy y a bx c d

andsc cc csyy x x

90 180 270 360-90-180-270-360

1

2

3

4

5

-1

-2

-3

-4

-5

x

y

• If a is positive, then there is no reflection about the x-axis.

• If a is negative, then there is a reflection about the x-axis.

• If b is positive, then there is no reflection about the y-axis.

• If b is negative, then there is a reflection about the y-axis.

Reflection:

Example : 2sin 2 1y x

-90-180

1

2

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4

5

-1

-2

-3

-4

-5

x

y

Example : 2sin 2 1y x

-90-180

1

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3

4

5

-1

-2

-3

-4

-5

x

y

2sin 2 1y x

-90-180

1

2

3

4

5

-1

-2

-3

-4

-5

x

y

2sin 2 1y x

-90-180

1

2

3

4

5

-1

-2

-3

-4

-5

x

y