Graphs of other Trig Functions

Graphs of other Trig Functions

Section 4.6

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

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”

Cosecant Curve

y = Csc x → y = Sin x

-1

1

2

23 2

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

Cosecant Curve

y = -2 Csc 4x + 2 → y = -2 Sin 4x + 2

2

4

8

4

83

2

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

Secant Curve

y = Sec 2x → y = Cos 2x

-1

1

4

2

43

Secant Curve

y = Sec x → y = Cos x

-1

1

2

23 2

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

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

-3

3

2 25

327 4

y = 2Sec (x + ) → y = 2 Cos (x + )

-2

2

2

2

23

2

2

y = ½ Csc (x - ) → y = ½ Csc (x - )

- ½

½

4

43

45

47

4

4

49

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

-2

2

21

83

41

81

Graph of Tangent and Cotangent

Still section 4.6

Tangent

Define tangent in terms of sine and cosine

Where is tangent undefined? xCos

Sin x

0 x CosWherever

23,

2 x

asymptotes

y = Tan x

2

02

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

y = Tan x

2

02

y =Tan xx

und.

2

und.2

00

4

4

-1 1

4

4

1

-1

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

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

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

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

y = 4Tan

y =4Tan x

und.

und.00

-4 42

22

x

2x

02

2

4

4

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

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

x und.

und.0π2

2

3

2 -24

34

5

4

5

43

2

23

2

2

y = 2Cot )2

( x

2Cot )2

( x

x und.

und.04

45

3 -32

43

2

4

45

3

3

y = 3 Cot )4

( x

3Cot )4

( x

43