Answer
The level curves:
$\theta = {\cos ^{ - 1}}z$, ${\ \ \ }$ for $0 \le \theta \le \pi $ and $ - 1 \le z \le 1$
Work Step by Step
In polar coordinates, we have $x = r\cos \theta $ and $y = r\sin \theta $.
So,
$f\left( {r,\theta } \right) = \frac{{r\cos \theta }}{r} = \cos \theta $
The level curves satisfy the equation
$\cos \theta = z$, ${\ \ \ }$ for $ - 1 \le z \le 1$
Thus, the level curves are
$\theta = {\cos ^{ - 1}}z$, ${\ \ \ }$ for $0 \le \theta \le \pi $
