The rectangular coordinates of a point are given. Express the point in polar coordinates with $r \geq 0 \text { and } 0 ^ { \circ } \leq \theta < 360 ^ { \circ } \text {. }$
- $( - \sqrt { 5 } , - \sqrt { 5 } )$

Question

The rectangular coordinates of a point are given. Express the point in polar coordinates with $$r \geq 0 \text { and } 0 ^ { \circ } \leq \theta < 360 ^ { \circ } \text {. }$$
-$( - \sqrt { 5 } , - \sqrt { 5 } )$
A) $\left( \sqrt { 5 } , 135 ^ { \circ } \right)$
B) $\left( \sqrt { 10 } , 315 ^ { \circ } \right)$
C) $\left( 10,225 ^ { \circ } \right)$
D) $\left( \sqrt { 10 } , 225 ^ { \circ } \right)$

Quizplus · Accepted Answer

To convert from rectangular coordinates $(x, y)$ to polar coordinates $(r, \theta)$, we use the formulas $r = \sqrt{x^2 + y^2}$ and $\theta = \tan^{-1}\left(\frac{y}{x}\right)$ (adjusting $\theta$ as necessary based on the quadrant). For the point $(- \sqrt { 5 } , - \sqrt { 5 })$, $r = \sqrt{(-\sqrt{5})^2 + (-\sqrt{5})^2} = \sqrt{10}$. Since both $x$ and $y$ are negative, the point is in the third quadrant, where $\theta = 180^\circ + \tan^{-1}\left(\frac{-\sqrt{5}}{-\sqrt{5}}\right) = 225^\circ$.

The Rectangular Coordinates of a Point Are Given r≥0 and 0∘≤θ<360∘. r \geq 0 \text { and } 0 ^ { \circ } \leq \theta < 360 ^ { \circ } \text {. }r≥0 and 0∘≤θ<360∘.

The Rectangular Coordinates of a Point Are Given $r \geq 0 \text { and } 0 ^ { \circ } \leq \theta < 360 ^ { \circ } \text {. }$