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

Step 1
To convert this rectangular coordinate ${\displaystyle (x,y)}$ to a polar coordinate ${\displaystyle (r,\theta )}$, use the following formulas:
${\displaystyle r^2=x^2+y^2}$
${\displaystyle \tan \theta =\frac{y}{x}}$
${\displaystyle r^2={(-200)}^2+{\left(10\right)}^2}$
${\displaystyle r^2=40100}$
${\displaystyle r=\sqrt{40100}}$
${\displaystyle r=10\sqrt{401}}$
${\displaystyle \tan \theta =\frac{y}{x}}$
${\displaystyle \tan \theta =\frac{10}{-200}}$
${\displaystyle \theta ={\tan }^{-1}\left(\frac{10}{-200}\right)}$
${\displaystyle \theta \approx -0.05}$
The angle -0.05 radians is in Quadrant IV, while the coordinate (-200, 10) is in Quadrant II. The angle is wrong because we used the ${\displaystyle \arctan }$ function, which only has a range of ${\displaystyle [-\frac{\pi }{2},\frac{\pi }{2}]}$
To find the correct angle, add ${\displaystyle \pi }$ to ${\displaystyle \theta }$
${\displaystyle -0.05+\pi =3.09}$
So, the polar coordinate is ${\displaystyle (10\sqrt{401},3.09)}$ or ${\displaystyle (200.25,3.09)}$