# What is the polar form of ( 5,14 )?

Jan 27, 2016

$\left(\sqrt{221} , 1.23\right)$

#### Explanation:

Using the formulae that link Cartesian to Polar coordinates.

• r^2 = x^2 + y^2

• theta = tan^-1 (y/x)

The point (5 , 14 ) is in the 1st quadrant and care must be taken

to ensure that $\theta \textcolor{b l a c k}{\text{ is in the 1st quadrant}}$

(here x = 5 and y = 14)

${r}^{2} = {5}^{2} + {14}^{2} = 25 + 196 = 221 \Rightarrow r = \sqrt{221}$

and $\theta = {\tan}^{-} 1 \left(\frac{14}{5}\right) = 1.23 \textcolor{b l a c k}{\text{ radians }}$

and $\theta \textcolor{b l a c k}{\text{ is in the 1st quadrant}}$

Polar coordinates are therefore (sqrt221 , 1.23 )

Jan 27, 2016

$\sqrt{221} \angle 70 , {3}^{\circ}$

#### Explanation:

Rectangular form $\left(x , y\right)$ can be converted into polar form $\left(r , \theta\right)$ as follows :

$r = \sqrt{{x}^{2} + {y}^{2}} \mathmr{and} \theta = {\tan}^{- 1} \left(\frac{y}{x}\right)$

So in this particular case we get

$r = \sqrt{{5}^{2} + {14}^{2}} = \sqrt{221}$

$\theta = {\tan}^{- 1} \left(\frac{14}{5}\right) = 70 , {3}^{\circ}$