# What does the polar equation theta=arcsin(1-3r) represent?

Nov 7, 2016

#### Explanation:

I is not clear what is exactly desired by the questioner.

However, $\theta = \arcsin \left(1 - 3 r\right)$ means

$\sin \theta = 1 - 3 r \Leftrightarrow r \sin \theta = r - 3 {r}^{2}$

or $3 r = 1 - \sin \theta$, which is a polar equation of a cardioid, a heart shaped figure, whose graph appears as one given below.

As polar coordinates $\left(r , \theta\right)$ are related to Cartesian coordinates $\left(x , y\right)$ by relations

$x = r \cos \theta$, $y = r \sin \theta$ and ${r}^{2} = {x}^{2} + {y}^{2}$

$\theta = \arcsin \left(1 - 3 r\right)$ is equivalent to

$y = \sqrt{{x}^{2} + {y}^{2}} - 3 \left({x}^{2} + {y}^{2}\right)$
graph{y=sqrt(x^2+y^2)-3(x^2+y^2) [-1.268, 1.232, -0.88, 0.37]}

Nov 7, 2016

$r = a \left(1 + \cos \left(\theta - \alpha\right)\right)$ is the equation representing a family of

cardioids through the pole r = 0. The parameter'a ' gives the size

and $\theta - \alpha$ is the line about which the cardioid is

symmetrical.

The period for $r \left(\theta\right)$ is $2 \pi$,

The given equation gives

$r = \frac{1}{3} \left(1 - \sin \theta\right) = \frac{1}{3} \left(1 + \cos \left(\theta + \frac{\pi}{2}\right)\right)$

This is the cardioid, with a = 1/3 and $\alpha = - \frac{\pi}{2}$

For making a graph, a short Table for the principal value range

$\left[- \frac{\pi}{2.} \frac{\pi}{2}\right]$ of arc sine is given below.

$\left(r , \theta\right) :$

(2/3, -pi/2) ((1/3(1+sqrt3/2), -pi/3) (1/2, -pi/6) (1/3, 0)

$\left(\frac{1}{6} , \frac{\pi}{6}\right) \left(\frac{1}{3} \left(1 - \frac{\sqrt{3}}{2}\right) , \frac{\pi}{3}\right) \left(0 , \frac{\pi}{2}\right)$

The conventional definition of arc sine restricts the graph to only a

range $\pi$, which is half of one period $2 \pi$.

For one full cardioid, continue the Table up to $\theta = \frac{3}{2} \pi$.

Note that, if the period $> \pi$, there are problems in adhering to

the principal value definition for arc sine. You could not get the full

cardioid.