Answer
No.
Work Step by Step
From the formulas
$r^{2}=x^{2}+y^{2}$ and $\displaystyle \tan\theta=\frac{y}{x}$,
finding $r^{2}$ leads to r having a possibility to be positive or negative (already more than one representation possible).
Solving $\displaystyle \tan\theta=\frac{y}{x}$ produces an angle $\theta$ as a valid angle in representing the point.
.
The angle$ \theta+\pi$ has the same tangent, so it could also be taken as a valid angle in representing the point.
$(r,\theta)$ and $(-r, \theta+\pi)$ represent the same point.
You could think of r as being the DIRECTED distance to the point.
If the terminal side of the angle passes through the point, r is positive,
and if it terminates in the symetrically opposite quadrant, r is negative.
Furthermore, for a given $\theta$, valid angles are also $\theta+2k\pi, k\in \mathbb{Z}$, so there are infinitely many polar coordinate representations for a single point.
Example 2 on page 589 has accompanying figures to demonstrate this.
So, no, the representation is not unique.
