Find three additional polar representations of the point $\left( - 5 , - \frac { 5 \pi } { 6 } \right)$, given in polar coordinates, using $- 2 \pi

Question

Find three additional polar representations of the point $\left( - 5 , - \frac { 5 \pi } { 6 } \right)$, given in polar coordinates, using $- 2 \pi < \theta < 2 \pi$.
A) $\left( - 5 , \frac { 7 \pi } { 6 } \right) , \left( - 5 , \frac { \pi } { 6 } \right) , \left( 5 , - \frac { 11 \pi } { 6 } \right)$
B) $\left( - 5 , \frac { 7 \pi } { 6 } \right) , \left( - 5 , \frac { \pi } { 6 } \right) , \left( - 5 , - \frac { 11 \pi } { 6 } \right)$
C) $\left( - 5 , \frac { 7 \pi } { 6 } \right) , \left( 5 , \frac { \pi } { 6 } \right) , \left( 5 , - \frac { 11 \pi } { 6 } \right)$
D) $\left( 5 , \frac { 7 \pi } { 6 } \right) , \left( 5 , \frac { \pi } { 6 } \right) , \left( - 5 , - \frac { 11 \pi } { 6 } \right)$
E) $\left( 5 , \frac { 7 \pi } { 6 } \right) , \left( - 5 , \frac { \pi } { 6 } \right) , \left( 5 , - \frac { 11 \pi } { 6 } \right)$

Quizplus · Accepted Answer

The correct additional representations consider the periodic nature of polar coordinates and the fact that changing the radius to its negative value flips the point to the opposite quadrant, requiring an adjustment of the angle by $\pi$ radians (or $180^\circ$). Option C correctly adjusts the angles and/or radius to represent the same point in the plane: $\left( - 5 , \frac { 7 \pi } { 6 } \right)$ flips the angle to the opposite quadrant without changing the radius, $\left( 5 , \frac { \pi } { 6 } \right)$ changes the radius to positive and adjusts the angle to a corresponding positive angle, and $\left( 5 , - \frac { 11 \pi } { 6 } \right)$ also uses a positive radius with an angle that correctly points to the same location in the plane.

Find Three Additional Polar Representations of the Point (−5,−5π6)\left( - 5 , - \frac { 5 \pi } { 6 } \right)(−5,−65π​)

Find Three Additional Polar Representations of the Point $\left( - 5 , - \frac { 5 \pi } { 6 } \right)$