Question

How to approach and solve this problem?

The curves with equations `x=asinnt` and `y=bcost` are called Lissajous figures . Investigate how these curves vary when `a`, `b`, and `n` vary. (Take `n` to be a positive integer.)

Answer 1

Here, for a=b,

for a.ne.b, the primary shape is elliptical
the primary shape is circular
the occurences of maximum values along x axis increase as the value of n increases,

for a>b, the primary shape is elliptical

Explanation:

`x=asinnt" and "y=bcost`

`x=acos(pi/2-nt), y=bsin(pi/2-t)`
`"represent an ellipse with a as major axis and b as minor axis"`

`"The initial orientation of the ellipse being vertical,"` `" major axis orienting along y axis while the"` `" minor axis orients along x axis"`

`"For n=1, "`

`x=acos(pi/2-t), y=bsin(pi/2-t)`

`"For n=2, "`

`x=acos(pi/2-2t), y=bsin(pi/2-t)`

`"For n=3, "`

`x=acos(pi/2-3t), y=bsin(pi/2-t)`

Here, for a=b,

for a.ne.b, the primary shape is elliptical
the primary shape is circular
the occurences of maximum values along x axis increase as the value of n increases,

for a>b, the primary shape is elliptical

t p/2-nt pi2-t cos(pi/2-t) sin(pi.2-t)
n=1
0.00 1.57 1.57 0.00 1.00
0.26 1.31 1.31 0.26 0.97
0.52 1.05 1.05 0.50 0.87
0.79 0.79 0.79 0.71 0.71
1.05 0.52 0.52 0.87 0.50
1.31 0.26 0.26 0.97 0.26
1.57 0.00 0.00 1.00 0.00
1.83 -0.26 -0.26 0.97 -0.26
2.09 -0.52 -0.52 0.87 -0.50
2.36 -0.79 -0.79 0.71 -0.71
2.62 -1.05 -1.05 0.50 -0.87
2.88 -1.31 -1.31 0.26 -0.97
3.14 -1.57 -1.57 0.00 -1.00
3.40 -1.83 -1.83 -0.26 -0.97
3.67 -2.09 -2.09 -0.50 -0.87
3.93 -2.36 -2.36 -0.71 -0.71
4.19 -2.62 -2.62 -0.87 -0.50
4.45 -2.88 -2.88 -0.97 -0.26
4.71 -3.14 -3.14 -1.00 0.00
4.97 -3.40 -3.40 -0.97 0.26
5.24 -3.67 -3.67 -0.87 0.50
5.50 -3.93 -3.93 -0.71 0.71
5.76 -4.19 -4.19 -0.50 0.87
6.02 -4.45 -4.45 -0.26 0.97
6.28 -4.71 -4.71 0.00 1.00
n=2
0.00 1.57 1.57 0.00 1.00
0.26 1.31 1.05 0.50 0.97
0.52 1.05 0.52 0.87 0.87
0.79 0.79 0.00 1.00 0.71
1.05 0.52 -0.52 0.87 0.50
1.31 0.26 -1.05 0.50 0.26
1.57 0.00 -1.57 0.00 0.00
1.83 -0.26 -2.09 -0.50 -0.26
2.09 -0.52 -2.62 -0.87 -0.50
2.36 -0.79 -3.14 -1.00 -0.71
2.62 -1.05 -3.67 -0.87 -0.87
2.88 -1.31 -4.19 -0.50 -0.97
3.14 -1.57 -4.71 0.00 -1.00
3.40 -1.83 -5.24 0.50 -0.97
3.67 -2.09 -5.76 0.87 -0.87
3.93 -2.36 -6.28 1.00 -0.71
4.19 -2.62 -6.81 0.87 -0.50
4.45 -2.88 -7.33 0.50 -0.26
4.71 -3.14 -7.85 0.00 0.00
4.97 -3.40 -8.38 -0.50 0.26
5.24 -3.67 -8.90 -0.87 0.50
5.50 -3.93 -9.42 -1.00 0.71
5.76 -4.19 -9.95 -0.87 0.87
6.02 -4.45 -10.47 -0.50 0.97
6.28 -4.71 -11.00 0.00 1.00
n=3
0.00 1.57 1.57 0.00 1.00
0.26 1.31 0.79 0.71 0.97
0.52 1.05 0.00 1.00 0.87
0.79 0.79 -0.79 0.71 0.71
1.05 0.52 -1.57 0.00 0.50
1.31 0.26 -2.36 -0.71 0.26
1.57 0.00 -3.14 -1.00 0.00
1.83 -0.26 -3.93 -0.71 -0.26
2.09 -0.52 -4.71 0.00 -0.50
2.36 -0.79 -5.50 0.71 -0.71
2.62 -1.05 -6.28 1.00 -0.87
2.88 -1.31 -7.07 0.71 -0.97
3.14 -1.57 -7.85 0.00 -1.00
3.40 -1.83 -8.64 -0.71 -0.97
3.67 -2.09 -9.42 -1.00 -0.87
3.93 -2.36 -10.21 -0.71 -0.71
4.19 -2.62 -11.00 0.00 -0.50
4.45 -2.88 -11.78 0.71 -0.26
4.71 -3.14 -12.57 1.00 0.00
4.97 -3.40 -13.35 0.71 0.26
5.24 -3.67 -14.14 0.00 0.50
5.50 -3.93 -14.92 -0.71 0.71
5.76 -4.19 -15.71 -1.00 0.87
6.02 -4.45 -16.49 -0.71 0.97
6.28 -4.71 -17.28 0.00 1.00
n=4
0.00 1.57 -4.71 0.00 1.00
0.26 1.31 -3.67 -0.87 0.97
0.52 1.05 -2.62 -0.87 0.87
0.79 0.79 -1.57 0.00 0.71
1.05 0.52 -0.52 0.87 0.50
1.31 0.26 0.52 0.87 0.26
1.57 0.00 1.57 0.00 0.00
1.83 -0.26 2.62 -0.87 -0.26
2.09 -0.52 3.67 -0.87 -0.50
2.36 -0.79 4.71 0.00 -0.71
2.62 -1.05 5.76 0.87 -0.87
2.88 -1.31 6.81 0.87 -0.97
3.14 -1.57 7.85 0.00 -1.00
3.40 -1.83 8.90 -0.87 -0.97
3.67 -2.09 9.95 -0.87 -0.87
3.93 -2.36 11.00 0.00 -0.71
4.19 -2.62 12.04 0.87 -0.50
4.45 -2.88 13.09 0.87 -0.26
4.71 -3.14 14.14 0.00 0.00
4.97 -3.40 15.18 -0.87 0.26
5.24 -3.67 16.23 -0.87 0.50
5.50 -3.93 17.28 0.00 0.71
5.76 -4.19 18.33 0.87 0.87
6.02 -4.45 19.37 0.87 0.97
6.28 -4.71 20.42 0.00 1.00