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

In polar coordinates, changing the sign of the radius $r$ flips the point across the origin, requiring an adjustment of $\pi$ radians (or 180 degrees) to the angle $\theta$ to point in the correct direction. The original point $\left( - 5 , - \frac { 5 \pi } { 6 } \right)$ can be represented in additional ways by either changing the sign of $r$ and adjusting $\theta$ accordingly or by adding multiples of $2\pi$ to $\theta$ (though the latter is restricted by the given range for $\theta$). The correct representations are:1. $\left( - 5 , \frac { 7 \pi } { 6 } \right)$ is incorrect because it simply adds $\pi$ to the angle without changing the sign of $r$, which is not a valid transformation given the original point.2. $\left( 5 , \frac { \pi } { 6 } \right)$ is correct because changing the sign of $r$ from $-5$ to $5$ and adjusting the angle by adding $\pi$ (to make the angle positive and correctly oriented) results in a valid representation.3. $\left( 5 , - \frac { 11 \pi } { 6 } \right)$ is correct because it represents the point after flipping across the origin (changing $r$ from $-5$ to $5$) and adjusting the angle to point in the same direction in the polar coordinate system.

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 $\left( - 5 , - \frac { 5 \pi } { 6 } \right)$