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

Oct 24, 2016

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

#### Explanation:

$r = r \left(\theta\right) = 3 + \sin 5 \theta$ is periodic with period $2 \frac{\pi}{5} = {72}^{o}$

Make the graph for one period $\theta \in \left[0 , {72}^{o}\right]$. The values of r

repeat in a cycle of period ${72}^{o}$.

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

$\left(r , {\theta}^{o}\right)$:

$\left(3 , 0\right) \left(3.707 , {9}^{o}\right) \left(4 , {18}^{o}\right) \left(3.707 , {27}^{o}\right) \left(3 , {36}^{o}\right) \left(2.283 , {45}^{o}\right)$

$\left(2 , {54}^{o}\right) \left(2.293 , {63}^{o}\right) \left(3 , {72}^{o}\right)$

The graph for $\theta \in \left[0 , 2 \pi\right]$ 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

$r = 3 + \cos 12 \theta$:
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]}