# Chapter 6 - Section 6.3 - Polar Coordinates - Exercise Set - Page 743: 46

The polar coordinates of $\left( -1,-\sqrt{3} \right)$ are $\left( 2,\frac{4\pi }{3} \right)$.

#### Work Step by Step

The polar coordinates of the point are $\left( r,\theta \right)$. Now, rewrite the polar coordinates in terms of rectangular coordinates as below: $r=\sqrt{{{x}^{2}}+{{y}^{2}}}$ …… (1) $\tan \theta =\frac{y}{x}$ …… (2) Substituting the values of $x\ \text{ and }\ y$ in (1) and (2), we get \begin{align} & r=\sqrt{{{x}^{2}}+{{y}^{2}}} \\ & =\sqrt{{{\left( -1 \right)}^{2}}+{{\left( -\sqrt{3} \right)}^{2}}} \\ & =\sqrt{1+3}=\sqrt{4} \\ & r=2 \end{align} And, \begin{align} & \tan \theta =\frac{y}{x} \\ & =\frac{-\sqrt{3}}{-1} \\ & \tan \theta =\sqrt{3} \end{align} Hence, $\tan \theta =\sqrt{3}$ And, $\tan \frac{\pi }{3}=\sqrt{3}$ Also, $\theta$ lies in the third quadrant which means \begin{align} & \theta =\pi +\frac{\pi }{3} \\ & \theta =\frac{4\pi }{3} \end{align} Therefore, the polar coordinates of $\left( -1,-\sqrt{3} \right)$ are $\left( 2,\frac{4\pi }{3} \right)$.

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