Question

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

Answer 1

Please see below.

Explanation:

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

However, `theta=arcsin(1-3r)` means

`sintheta=1-3rhArrrsintheta=r-3r^2`

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

As polar coordinates `(r,theta)` are related to Cartesian coordinates `(x,y)` by relations

`x=rcostheta`, `y=rsintheta` and `r^2=x^2+y^2`

`theta=arcsin(1-3r)` is equivalent to

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

Answer 2

`r=a(1+cos(theta-alpha))` 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(theta)` is `2pi`,

The given equation gives

`r =1/3(1-sin theta)=1/3(1+cos(theta + pi/2))`

This is the cardioid, with a = 1/3 and `alpha=-pi/2`

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

`[-pi/2. pi/2]` of arc sine is given below.

`(r, theta):`

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

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

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

range `pi`, which is half of one period `2pi`.

For one full cardioid, continue the Table up to `theta = 3/2pi`.

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.