$\text { Find the polar coordinates, } 0 \leq \theta

Question

$\text { Find the polar coordinates, } 0 \leq \theta < 2 \pi \text { and } r \geq 0 \text {, of the point given in Cartesian coordinates. }$
-(4, -4)
A) A) $\left( 4 \sqrt { 2 } , \frac { \pi } { 4 } \right)$
B) $\left( 4 \sqrt { 2 } , \frac { 7 \pi } { 4 } \right)$
C) $\left( 4 \sqrt { 2 } , \frac { 5 \pi } { 4 } \right)$
D) $\left( 4 \sqrt { 2 } , \frac { 3 \pi } { 4 } \right)$

Quizplus · Accepted Answer

The polar coordinates $r$ and $\theta$ can be found from the Cartesian coordinates $(x, y)$ using $r = \sqrt{x^2 + y^2}$ and $\theta = \tan^{-1}\left(\frac{y}{x}\right)$. For the point (4, -4), $r = \sqrt{4^2 + (-4)^2} = \sqrt{32} = 4\sqrt{2}$. The angle $\theta$ in this case is in the fourth quadrant, where $\tan^{-1}\left(\frac{-4}{4}\right) = \tan^{-1}(-1)$, which corresponds to an angle of $\frac{7\pi}{4}$ radians when considering the full circle $0 \leq \theta < 2\pi$.