Question

A point has polar coordinates (r,Ф) and rectangular coordinates (x,y). If you know r and y, then:

a) it is not possible to calculate the polar or Cartesian coordinates
b) possible to get the polar coordinates but not the Cartesian coordinates
c) possible to get Cartesian coordinates, but not polar coordinates
d) possible to get Cartesian coordinates and polar coordinates

Answer 1

well, let's say we know "r" and also "y", can we get the angle and the rectangular "x"?

$\bf \begin{cases} y=rsin(\theta )\\ x=rcos(\theta ) \end{cases} \\\\[-0.35em] ~\dotfill\\\\ y=rsin(\theta )\implies \cfrac{y}{r}=sin(\theta )\implies sin^{-1}\left( \cfrac{y}{r} \right)=\theta$

and once we know what angle θ is, since x = rcos(θ), we know "r" already, and we know θ as well, so we know "x" as well.