Question

What kind of a polar graph is `r=19sin2x`?

Answer 1

Graph is a continuous finite curve which looks like two petals in first and third quadrant.

Explanation:

Let us convert the polar graph is `r=19sin2x` (we are assuming `theta` has been written as `x` in this equation) in Cartesian coordinates `(x,y)`. As relation between two is given by `r^2=x^2+y^2`, `rcostheta=x` and `rsintheta=y`. Let us eliminate `r` and `theta` using these,

`r=19sin2theta` can be written as `r=19*2sinthetacostheta` or

`r=19*2*y/r*x/r` or `r^3=38xy` or `(r^2)^1.5=38xy`

`graph{(x^2+y^2)^1.5=38xy [-18.02, 18.03, -9.01, 9.01]}`

As is seen, graph is a continuous finite curve which looks like two petals in first and third quadrant.