Ecological Succession. ( ECOSYSTEM, B. Pharmacy, 1st Year, Sem-II, Environmen...
8 points on the unit circle the wrapping function w(t)
1. Objectives: BTEOTPSWBAT
• Find points on the Unit Circle.
• Use the Wrapping Function W(t) to find points
(x, y) on the Unit Circle.
2. Warm-up:
Find a point on the unit circle at π .
3
The Unit Circle has a radius of 1 unit (r = 1).
1 3
,
÷
2 2 ÷
π
3
1
60°
1
2
3
2
3. Find a point on the unit circle at 3π .
4
2 2
,
−
÷
2 2 ÷
3π
4
2
2
1
45°
2
−
2
4.
5.
6. Find a point on the unit circle at 11π .
6
3 1
,− ÷
2
2÷
3
2
30°
1
1
−
2
11π
6
7. Find a point on the unit circle at π .
6
3 1
, ÷
2 2÷
π
6
1
30°
3
2
1
2
8. Find a point on the unit circle at 2π .
3
1 3
− ,
÷
2 2 ÷
2π
3
3
2
1
60°
−
1
2
9. Find a point on the unit circle at 7π .
4
2
2
,−
÷
2
2 ÷
2
2
45°
1
7π
4
2
−
2
10. Find a point on the unit circle at 10π
.
−
1 3
− ,
÷
2 2 ÷
−10π
3
3
2
1
60°
1
−
2
3
11. Objectives: BTEOTPSWBAT
• Find points on the Unit Circle.
• Use the Wrapping Function W(t) to find points
(x, y) on the Unit Circle.
12. Now “wrap” the number line around the circle.
Each real number on the number line corresponds to a point (x, y)
on the unit circle (r = 1).
t
13. Name the point where W(t) is located.
1) W (π ) = (−1, 0)
2) W (4π ) = (1, 0)
π
3) W − ÷= (0, −1)
2
5π
4) W ÷ = (0,1)
2
5) W ( −3π ) = (−1, 0)
14. Name the quadrant where W(t) is located.
π
I
1) W ÷
3
7π
2) W
÷ II
8
2π
3) W −
÷ III
3
4) W ( 3)
II
III
5) W (−3)
2π
6) W 3π +
÷ IV
3
54π II
7) W
54
÷
*Note: This is different than W ÷, which would be in I.
8
8
15. How can you tell if a point is on the Unit Circle?
3 4
Is the point , ÷ on the Unit Circle?
5 5
2
2
3 4
2
Does ÷ + ÷ = 1 ?
5 5
9 16
+
=1
25 25
25
=1
25
16. 3 4
Given W : t → , ÷ , find each of the following:
5 5
3 4
3 4
Note: W : t → , ÷ is the same as W(t)= , ÷.
5 5
5 5
4
3
1) W (−t ) = , − ÷
5
5
3 4
2) W (2π + t ) = , ÷
5 5
3 4
3) W ( 4π + t ) = , ÷
5 5
17. 3 4
Given W : t → , ÷ , find each of the following:
5 5
4
3
4) W ( π + t ) = − , − ÷
5
5
3 4
5) W (π − t ) = − , ÷
5 5
4
3
6) W (t − π ) = − , − ÷
5
5
18. 3 1
Prove that the point
, ÷ is on the Unit Circle?
2 2
2
3 1
2
÷ + ÷ =1
2 2
2
3 1
+ =1
4 4
4
=1
4
19. Find each of the following points on the Unit Circle:
3 1
π
1) W : → =
, ÷
2 2÷
6
5π
3 1
2) W :
→ = −
, ÷
2 2÷
6
3
1
7π
3) W
, − ÷
÷= −
2÷
6 2
1
13π 3
4) W −
÷=
2 , − 2÷
÷
6
