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
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