Can we convert polar to rectangular when we are given $(1,\theta )$ where $ r=1$ and $0\le \theta <2\pi $?

Question

Can we convert polar to rectangular when we are given $(1,\theta )$ where $ r=1$ and $0\le \theta <2\pi $?

Answer 1

When in doubt plot it. It might help you see the answer.

The basic rules for converting between caretesian and polar.

$x = r \cos \theta\\ y = r \sin \theta$

$\theta = \tan^{-1} (\frac yx)\\ r = \sqrt{x^2 + y^2}$

So what do you know?

$r = 1.$ What does that mean for $(x,y)$?

Answer 2

$(r, \theta) = (1, \theta)$ geometrically means that the distance from the origin is [constant] 1, and the angle can vary freely. One particular shape should come to mind. Or as Doug M points out, in this case you obtain a familiar parametrisation $(x,y) = (\cos \theta, \sin \theta)$.