Find the polar coordinates of the point $( - 2 \sqrt { 3 } , 2 )$ given in rectangular coordinates for $\theta \in [ 0,2 \pi ]$ and $r \geq 0$
A) $\left( 4 , \frac { \pi } { 6 } \right)$
B) $\left( 4 , \frac { \pi } { 3 } \right)$
C) $\left( 4 , \frac { 2 \pi } { 3 } \right)$
D) $\left( 4 , \frac { 5 \pi } { 6 } \right)$

Question

Quizplus · Accepted Answer

The polar coordinates can be found using $r = \sqrt{x^2 + y^2}$ and $\theta = \tan^{-1}\left(\frac{y}{x}\right)$, where $x = -2\sqrt{3}$ and $y = 2$. Calculating $r$ gives $r = \sqrt{(-2\sqrt{3})^2 + 2^2} = 4$. For $\theta$, since the point is in the second quadrant, $\theta = \pi - \tan^{-1}\left(\frac{2}{-2\sqrt{3}}\right) = \pi - \tan^{-1}\left(\frac{1}{-\sqrt{3}}\right) = \pi - \left(-\frac{\pi}{6}\right) = \frac{5\pi}{6}$.

Find the Polar Coordinates of the Point (−23,2)( - 2 \sqrt { 3 } , 2 )(−23​,2)

Find the Polar Coordinates of the Point $( - 2 \sqrt { 3 } , 2 )$