# What is the polar form of (-200,10)?

Jul 15, 2017

$\left(10 \sqrt{401} , 3.09\right)$

#### Explanation:

To convert this rectangular coordinate $\left(x , y\right)$ to a polar coordinate $\left(r , \theta\right)$, use the following formulas:

${r}^{2} = {x}^{2} + {y}^{2}$
$\tan \theta = \frac{y}{x}$

${r}^{2} = {\left(- 200\right)}^{2} + {\left(10\right)}^{2}$
${r}^{2} = 40100$
$r = \sqrt{40100}$
$r = 10 \sqrt{401}$

$\tan \theta = \frac{y}{x}$

$\tan \theta = \frac{10}{-} 200$
$\theta = {\tan}^{-} 1 \left(\frac{10}{-} 200\right)$
$\theta \approx - 0.05$

The angle $- 0.05$ radians is in Quadrant $I V$, while the coordinate $\left(- 200 , 10\right)$ is in Quadrant $I I$. The angle is wrong because we used the $\arctan$ function, which only has a range of $\left[- \frac{\pi}{2} , \frac{\pi}{2}\right]$. To find the correct angle, add $\pi$ to $\theta$.

$- 0.05 + \pi = 3.09$

So, the polar coordinate is $\left(10 \sqrt{401} , 3.09\right)$ or $\left(200.25 , 3.09\right)$.