# How do you convert (3, - 4)  into polar coordinates?

Jan 4, 2016

$\left(r , \theta\right) = \left(5 , 2 \pi + \arctan \left(- \frac{3}{4}\right)\right)$ (approximately $= \left(5 , 5.64\right)$

#### Explanation:

The radius $r$ is given by the Pythagorean Theorem as
$\textcolor{w h i t e}{\text{XXX}} r = \sqrt{{\left(- 4\right)}^{2} + {3}^{2}} = 5$

The value of $\theta$ is based on the $\arctan \left(\frac{y}{x}\right)$ but is dependent upon the quadrant in which $\left(x , y\right)$ is positioned. This is because the $\arctan$ function returns a value in the range $\left[- \frac{\pi}{2} , + \frac{\pi}{2}\right]$ relative to the X-axis.

{: (color(black)("Quadrant"),,color(white)("XXX")color(black)(theta)), (bar(color(white)("XXXXXX")),,bar(color(white)("XXXXXX"))), (I,,arctan(y/x)), (II,,arctan(y/x)+pi), (III,,arctan(y/x)+pi), (IV,,arctan(y/x)+2pi) :}