Mathematics Dictionary
Dr. K. G. Shih
Graphs of R = a + b*F(p*A/q)^M
Subjects
Symbol Defintion
Example : x^2 measn square of x
AN 10 01 |
- Sketch graphs of R = a + b*sin(p*A/q)^M
AN 10 02 |
- Sketch graphs of R = a + b*sin(p*A/q)^M
AN 10 03 |
- Patterns
AN 10 04 |
- Petals of R = cos(2*A)
AN 10 05 |
- Petals of R = cos(3*A)
AN 10 06 |
- Petals of R = sin(3*A/2)
AN 10 07 |
- Petals of R = cos(3*A/2)
AN 10 08 |
- Twin patterns of R = sin(p*A/q) and R = cos(p*A/q)
AN 10 09 |
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AN 10 10 |
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Answers
AN 10 01. Sketch graphs of R = a + b*sin(p*A/q)^M
Sketch
Graphs in Analytic geometry
Section 10 in anlytic geometry
Examples
10 01 Graph of R = 1 + 1*sin(11*A/4)^3
Select section 10 in upper box
Select program 01 in lower box
Give a,b,p,q,and M : 1,1,11,4,3
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AN 10 02. Graphs of R = a + b*cos(p*A/q)^M
Sketch
Graphs in Analytic geometry
Section 10 in anlytic geometry
Examples
10 02 Graph of R = 1 + 1*cos(11*A/4)^3
Select section 10 in upper box
Select program 02 in lower box
Give a,b,p,q,and M : 1,1,11,4,3
Question : What is the difference of R=1+1*sin(11*A/4)^3 and R=1+1*sin(11*A/4)^3 ?
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AN 10 03. Patterns
Demo Patterns
Patterns
Topics
1. Introduction
2. Patterns of R = sin(p*A/q)^M
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AN 10 04. Petals of R = cos(2*A)
Find direction of petals
Number 3*A ...... Angle A ... Value R ... Plot angle ... (Direction of Petal)
1 .... 000 ...... 000 ....... +1 ........ 000
2 .... 180 ...... 090 ....... -1 ........ 270 .......... (090 + 180) = 270
3 .... 360 ...... 180 ....... +1 ........ 180
4 .... 540 ...... 270 ....... -1 ........ 090 .......... 270 + 180 - 360 = 90
5 .... 720 ...... 360 ....... +1 ........ 000
Find number of petals and domain
Since number 5 petal and number 1 petal have same direction
Hence it has 4 petals
The domain is 360 - 0 = 360
Hence A = (0, 360)
Compare the direction of petals of R = cos(2*A) and R = sin(2*A)
R = cos(2*A) 000 090 180 270 (TR 10 06)
R = sin(2*A) 045 135 225 315(TR 10 03)
Hence the petals have 45 degrees differenct
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AN 10 05. Petals of R = cos(3*A)
Find direction of petals
Number 3*A ...... Angle A ... Value R ... Plot angle ... (Direction of Petal)
1 .... 000 ...... 000 ....... +1 ........ 000
2 .... 180 ...... 060 ....... -1 ........ 240 .......... (060 + 180) = 240
3 .... 360 ...... 120 ....... +1 ........ 120
4 .... 540 ...... 180 ....... -1 ........ 000 .......... 180 + 180 - 360 = 0
Find number of petals and domain
Since number 4 petal and number 1 petal have same direction
Hence it has 3 petals
The domain is 180 - 00 = 180
Hence A = (0, 180)
Compare the direction of petals of R = cos(3*A) and R = sin(3*A)
R = cos(3*A) 000 120 240 (TR 10 05)
R = sin(3*A) 030 150 270 (TR 10 03)
Hence the petals have 30 degrees differenct
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AN 10 06. Petals of R = sin(3*A/2)
Find direction of petals
Number 3*A/2 .... Angle A ... Value R ... Plot angle ... (Direction of Petal)
1 .... 090 ...... 060 ....... +1 ........ 060
2 .... 270 ...... 180 ....... -1 ........ 000 .......... (180 + 180) = 260
3 .... 450 ...... 300 ....... +1 ........ 300
4 .... 630 ...... 420 ....... -1 ........ 240 .......... 420 + 180 - 360 = 240
5 .... 810 ...... 540 ....... +1 ........ 180 .......... 540 - 360 = 180
6 .... 990 ...... 660 ....... -1 ........ 120 .......... 660 + 180 - 720 = 120
7 ... 1170 ...... 780 ....... +1 ........ 060 .......... 780 - 720 = 060
Find number of petals and domain
Since number 7 petal and number 1 petal have same direction
Hence it has 6 petals
The domain is 780 - 60 = 720
Hence A = (0, 720)
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AN 10 07. Petals of R = cos(3*A/2)
Find direction of petals
Number 3*A/2 .... Angle A ... Value R ... Plot angle ... (Direction of Petal)
1 .... 000 ...... 000 ....... +1 ........ 000
2 .... 180 ...... 120 ....... -1 ........ 300 .......... (120 + 180) = 300
3 .... 360 ...... 240 ....... +1 ........ 240
4 .... 540 ...... 360 ....... -1 ........ 180 .......... 360 + 180 - 360 = 180
5 .... 720 ...... 480 ....... +1 ........ 300 .......... 480 + 180 - 360 = 300
6 .... 900 ...... 600 ....... -1 ........ 060 .......... 600 + 180 - 720 = 060
7 ... 1080 ...... 720 ....... +1 ........ 000 .......... 720 - 720 = 000
Find number of petals and domain
Since number 7 petal and number 1 petal have same direction
Hence it has 6 petals
The domain is 720 - 0 = 720
Hence A = (0, 720)
Compare petal direction of R = sin(3*A/2) and R = cos(3*A/2)
Comparison
R = cos(3*A/2) 000 060 120 180 240 300 (TR 10 07)
R = sin(3*A/2) 000 060 120 180 240 300 (TR 10 06)
Conclusion
From AN 10 06 and AN 10 07
We see that all the directions of R = cos(3*A/2) are same as R = sin(3*A/2)
From the final diagram, we can not distinguish the graphs
They are congruent with same direction
We then define it as twin patterns (See Pattern Mathematics by Dr. shih)
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AN 10 08. Twin patterns of R = sin(p*A/q) and R = cos(p*A/q)
Definition of twin patterns
Graphs of R = sin(p*A/q) and R = cos(p*A/q) are congruent
Also they have same direction (No phase difference)
Conditions
The value of p is odd
The value of q is even
The cycle domain is A = 0 to 2*q*pi
Examples
See AN 10 06 and AN 10 07
Sketch diagrams
Graphs in Analytic geometry
Section 21 10
Use 21 10 to enter the program
Example 1 : Sketch R = sin(3*A/2)^1
Click Menu
Click 10 in upper box
Click 01 in lower box
Give data of R = a + b*sin(p*A/q)^M : 0,1,3,2,1
Example 2 : Sketch R = cos(3*A/2)^1
Click Menu
Click 10 in upper box
Click 02 in lower box
Give data of R = a + b*sin(p*A/q)^M : 0,1,3,2,1
Demo diagrams
Pattern Mathematics
Section 16 01
Click start
Click 16 in upper box
Click 01 in lower box
Give data : 2,1
Reference
Pattern Mathematics published by Dr. K. G. Shih
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AN 10 09. Answer
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AN 10 10. Answer
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