Question

How do you graph `r=3+sin(5theta)`?

Answer 1

See 5 loops of `r = sin 5theta` transformed to 5 waves, twining around the circle r = 3, in the graph of `r = 3 + sin 5theta`.

Explanation:

`r = r(theta)=3+sin 5theta` is periodic with period `2pi/5=72^o`

Make the graph for one period `theta in [0, 72^o]`. The values of r

repeat in a cycle of period `72^o`.

Use this Table to get the wave-like curve for one period. ..

`(r, theta^o)`:

`(3, 0) (3.707, 9^o) (4, 18^o) (3.707, 27^o) (3, 36^o) (2.283, 45^o)`

`(2, 54^o) (2.293, 63^o) (3, 72^o)`

The graph for `theta in [0, 2pi]` comprises 5 full waves twining

around the circle r = 3.

Indeed, interesting.

Analogy: The path of either pole on the Earth's surface with a

compound period of about 256 centuries, for its precession, is

similar ( but complicated) on the surface of our Earth. The radius of

this circle is about 2535 km.

Graph of #r = 3 + sin 5theta and the circle around which this

twines#:
graph{((x^2+y^2)^3-3(x^2+y^2)^2.5-5 x^4y+10x^2y^3-y^5)(x^2+y^2-9)=0 [-8 8 -4 4]}

Graph of 12 waves twining around a circle in

`r = 3 + cos 12theta`:
graph{((x^2+y^2)^6.5-3(x^2+y^2)^6-12 (x^11y-xy^11)+220(x^9y^3-x^3y^9)-792(x^7y^5-x^5y^7))(x^2+y^2-9)=0 [-8 8 -4.2 4.2]}