4.2 the wrapping function
19
Chapter 4.2 The Wrapping Function 1
-
Upload
lester-abando -
Category
Education
-
view
52 -
download
0
Transcript of 4.2 the wrapping function
Chapter 4.2 The Wrapping Function
1
-5 -4 -3 -2 -1 1 2 3 4 5
-5
-4
-3
-2
-1
1
2
3
4
5
1
0
2
1
2
2
-5 -4 -3 -2 -1 1 2 3 4 5
-5
-4
-3
-2
-1
1
2
3
4
5
0
1
2
12
3
3
-5 -4 -3 -2 -1 1 2 3 4 5
-5
-4
-3
-2
-1
1
2
3
4
5
1
0
2
12
3
4
-5 -4 -3 -2 -1 1 2 3 4 5
-5
-4
-3
-2
-1
1
2
3
4
5
Each real number corresponds
to a length of an arc whose initial
point is 1,0 .
In counterclockwise direction,
the arc is positive.
In clockwise direction, the arc
is negative.
0
2
12
3
1
5
Wrapping Function
The is a function from
the set of real numbers to the set of all points
on the unit circle with center at the origin
such that
,
where , is the terminal p
wrapping fu
oint of the
n
arc
with leng
ction P
P t x y
x y
th and whose initial point is 1,0 .t
6
-5 -4 -3 -2 -1 1 2 3 4 5
-5
-4
-3
-2
-1
1
2
3
4
5
,P t x y
t
,x y
7
-5 -4 -3 -2 -1 1 2 3 4 5
-5
-4
-3
-2
-1
1
2
3
4
5Example 4.2.1
Evaluate the following.
1. 0,12
2. 2 1,0
3. 0 1,0
4. 1,0
P
P
P
P
2
2
3
2
8
-5 -4 -3 -2 -1 1 2 3 4 5
-5
-4
-3
-2
-1
1
2
3
4
5
5. 0, 12
6. 1,0
P
P
2
2
3
2
9
Special Arc Lengths
2
32
2
43
4
54
74
2 2,
4 2 2P
3 2 2,
4 2 2P
5 2 2,
4 2 2P
7 2 2,
4 2 2P
10
Special Arc Lengths
2
32
3
43
53
1 3,
3 2 2P
2 1 3,
3 2 2P
4 1 3,
3 2 2P
5 1 3,
3 2 2P
11
Special Arc Lengths
2
32
3
43
53
6
2
56
76
116
32
12
Special Arc Lengths
3 1,
6 2 2P
5 3 1,
6 2 2P
7 3 1,
6 2 2P
11 3 1,
6 2 2P
65
6
76
116
13
Challenge!
Evaluate the following.
5 2 2 2 1 31. , 5. ,
4 2 2 3 2 2
11 3 1 11 3 12. , 6. ,
6 2 2 6 2 2
1 3 1 33. , 7. ,
3 2 2 3 2 2
34.
4
P P
P P
P P
P
2 2 7 3 1, 8. ,
2 2 6 2 2P
14
Coterminal Arcs
Two arcs with the same terminal p
c
o
o
in
te
ts
ar rmine called al a rcs.
15
4 7
4 9
4
7 9, , and are coterminal arcs.
4 4 4
16
Coterminal Arcs
Coterminal arcs have equal wrapping
function values.
17
Example 4.2.2
Evaluate the following.
1. 3 1,0
2. 6 0 1,0
7 1 33. ,
3 3 2 2
101 5 2 24. ,
4 4 2 2
23 11 3 15. ,
6 6 2 2
P P
P P
P P
P P
P P
18
End of Chapter 4.2
19
