A curve is described in polar coordinates . Find parametric equations for $x$ and $y$ and plot the curve.

Question

A curve is described in polar coordinates by the equations $$ r = t; \theta = 3 \cos t; 0 ≤ t ≤ 10 $$ Find parametric equations for $x$ and $y.$

I cannot convert it into parametric form

Answer 1

You already have it in parametric form with $t$ as a parameter. Ignoring the multivalue problem of the cosine for a moment, you have $\sqrt {x^2+y^2}=t, \arctan \frac yx= 3 \cos t$, two equations in two unknowns. You need to solve these for $x$ and $y$, which will be functions of $t$ as desired.

Answer 2

the transformation between cartesian $(x, y)$ and the polar $[r, \theta]$ are given by $$x = r \cos \theta, y = \sin \theta $$ so the parametric equation in cartesian is $$x = t \cos (3 \cos t), y = t \sin (3 \cos t),0 \le t \le 10. $$

the curve follows a spiral bouncing back and forth between the rays $\theta = -3$ and $\theta = 3$ while getting farther and farther.