Download - Graphs of other Trig Functions

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Page 1: Graphs of other  Trig Functions

Graphs of other Trig Functions

Section 4.6

Page 2: Graphs of other  Trig Functions

Cosecant Curve

What is the cosecant x?

Where is cosecant not defined?◦Any place that the Sin x = 0

The curve will not pass through these points on the x-axis.

Sin x1

x = 0, π, 2 π

Page 3: Graphs of other  Trig Functions

Cosecant Curve

Drawing the cosecant curve

1) Draw the reciprocal curve2) Add vertical asymptotes wherever curve

goes through horizontal axis3) “Hills” become “Valleys” and

“Valleys” become “Hills”

Page 4: Graphs of other  Trig Functions

Cosecant Curve

y = Csc x → y = Sin x

-1

1

2

23 2

Page 5: Graphs of other  Trig Functions

Cosecant Curve

y = 3 Csc (4x – π) → y = 3 Sin (4x – π)a = 3 b = 4

Per. = 2

dis. = 8

c = π P.S. = 4

-3

3

4

83

2

85

43

Page 6: Graphs of other  Trig Functions

Cosecant Curve

y = -2 Csc 4x + 2 → y = -2 Sin 4x + 2

2

4

8

4

83

2

Page 7: Graphs of other  Trig Functions

Secant Curve

What is the secant x?

Where is secant not defined?◦Any place that the Cos x = 0

The curve will not pass through these points on the x-axis.

xCos1

23

2

Page 8: Graphs of other  Trig Functions

Secant Curve

y = Sec 2x → y = Cos 2x

-1

1

4

2

43

Page 9: Graphs of other  Trig Functions

Secant Curve

y = Sec x → y = Cos x

-1

1

2

23 2

Page 10: Graphs of other  Trig Functions

Graph these curves

1) y = 3 Csc (πx – 2π)

2) y = 2 Sec (x + )

3) y = ½ Csc (x - )

4) y = -2 Sec (4x + 2)

42

Page 11: Graphs of other  Trig Functions

y = 3Csc (πx – 2π) → y = 3 Sin (π x – 2π)

-3

3

2 25

327 4

Page 12: Graphs of other  Trig Functions

y = 2Sec (x + ) → y = 2 Cos (x + )

-2

2

2

2

23

2

2

Page 13: Graphs of other  Trig Functions

y = ½ Csc (x - ) → y = ½ Csc (x - )

- ½

½

4

43

45

47

4

4

49

Page 14: Graphs of other  Trig Functions

y = -2 Sec (4π x + 2 π) -2 Cos (4π x + 2 π)

-2

2

21

83

41

81

Page 15: Graphs of other  Trig Functions

Graph of Tangent and Cotangent

Still section 4.6

Page 16: Graphs of other  Trig Functions

Tangent

Define tangent in terms of sine and cosine

Where is tangent undefined? xCos

Sin x

0 x CosWherever

23,

2 x

asymptotes

Page 17: Graphs of other  Trig Functions

y = Tan x

2

02

Page 18: Graphs of other  Trig Functions

Tangent Curve

So far, we have the curve and 3 key points

Last two key points come from the midpoints between our asymptotes and the midpoint◦Between and 0 and between and 0

→ and

2

2

4

4

Page 19: Graphs of other  Trig Functions

y = Tan x

2

02

y =Tan xx

und.

2

und.2

00

4

4

-1 1

4

4

1

-1

Page 20: Graphs of other  Trig Functions

For variations of the tangent curve

1) Asymptotes are found by using:

A1. bx – c = A2. bx – c =

2) Midpt. =

3) Key Pts: and

2

2

2A2 A1

2Midpt A1

2Midpt A2

Page 21: Graphs of other  Trig Functions

y = 2Tan 2x

y =2Tan 2xx

und.

und.4

4

4

4

bx – c = 2

bx – c = 2

2x= 2

2x = 2

x = 4

x = 4

Page 22: Graphs of other  Trig Functions

y = 2Tan 2x

0

y =2Tan 2xx

und.

und.004

4

-2 2

4

4

Midpt = 24

4

K.P. = = 2

0 4

8

K.P. = = 2

0 4

8

8

8

8

8

= 0

Page 23: Graphs of other  Trig Functions

y = 4Tan

y =4Tan x

und.

und.00

-4 42

22

x

2x

2

cbx2

cbx

22

x

x22

x

x

221 AAMidpt

2

Midpt

0Midpt

21.. MidptAPK

20..

PK

2..

PK

22.. MidptAPK

20..

PK

2.. PK

Page 24: Graphs of other  Trig Functions

y = 4Tan

y =4Tan x

und.

und.00

-4 42

22

x

2x

02

2

4

4

Page 25: Graphs of other  Trig Functions

Cotangent Curve

Cotangent curve is very similar to the tangent curve. Only difference is asymptotes

bx – c = 0 bx – c = π

→ 0 and π are where Cot is undefined

Page 26: Graphs of other  Trig Functions

y = 2Cotx

und.

und.0π2

2

3

2 -24

34

5)

2(

x

0 cbx cbx

02

x

2

x

2x

23

x

221 AAMidpt

22

32

Midpt

Midpt

21.. MidptAPK

22..

PK

43..

PK

22.. MidptAPK

22

3

..

PK

45..

PK

2Cot )2

( x

Page 27: Graphs of other  Trig Functions

x und.

und.0π2

2

3

2 -24

34

5

4

5

43

2

23

2

2

y = 2Cot )2

( x

2Cot )2

( x

Page 28: Graphs of other  Trig Functions

x und.

und.04

45

3 -32

43

2

4

45

3

3

y = 3 Cot )4

( x

3Cot )4

( x

43