Question

Which polar coordinates represent the same point as the rectangular coordinate (2.-1) ? Select all that apply.

Answer 1

Solution:

Given:

The rectangular coordinate;

$\begin{gathered} (2,-1) \\ \\ where: \\ x=2 \\ y=-1 \end{gathered}$

To convert to polar coordinate;

$\begin{gathered} (r,\theta) \\ \\ where: \\ r=\sqrt{x^2+y^2} \\ \theta=tan^{-1}(\frac{y}{x}) \end{gathered}$

Hence,

$\begin{gathered} r=\sqrt{2^2+(-1)^2} \\ r=\sqrt{4+1} \\ r=\sqrt{5} \\ \\ \\ \theta=tan^{-1}(-\frac{1}{2}) \\ \theta=-26.565 \\ \theta=-26.6^0 \\ tan\text{ is negative in the second and fourth quadrants} \\ Hence, \\ \\ In\text{ the second quadrant;} \\ \theta=180-26.6 \\ \theta=153.4^0 \\ \\ In\text{ the fourth quadrant;} \\ \theta=360-26.6 \\ \theta=333.4^0 \end{gathered}$

Therefore, as a polar coordinate, (2,-1) is;

$(\sqrt{5},333.4^0)\text{ OR }(-\sqrt{5},153.4^0)$

Thus, the correct answers are OPTION A and OPTION C.