# How do you find all polar coordinates of point P where P =(9, (2pi)/3)?

Jan 26, 2018

Polar coordinates of $\textcolor{g r e e n}{P \left(9.2405 , {13.1}^{0}\right)}$

#### Explanation:

$P \left(9 , \frac{2 \pi}{3}\right)$

$x = r \cos \theta , y = r \sin \theta$

Hence $r = \sqrt{{x}^{2} + {y}^{2}} , = \sqrt{{9}^{2} + {\left(\frac{2 \pi}{3}\right)}^{2}} = 9.2405$

$\tan \theta = \frac{y}{x} = \frac{\frac{2 \pi}{3}}{9} = \frac{2 \pi}{27}$

$\theta = {\tan}^{- 1} \left(\frac{2 \pi}{27}\right) = {13.1}^{0}$

Polar coordinates of $\textcolor{g r e e n}{P \left(9.2405 , {13.1}^{0}\right)}$

Jan 26, 2018

$\left(9 , \frac{2 \pi}{3} + 2 \pi n\right)$ or $\left(- 9 , \frac{5 \pi}{3} + 2 \pi n\right)$, $n \in \mathbb{Z}$.

Or in one shot: $\left({\left(- 1\right)}^{n} \cdot 9 , \frac{2 \pi}{3} + \pi \cdot n\right)$, $n \in \mathbb{Z}$

#### Explanation:

Given $\left(9 , \frac{2 \pi}{3}\right)$, any angle coterminal to $\frac{2 \pi}{3}$ will put us in the same place, so:

$\left(9 , \frac{2 \pi}{3} + 2 \pi n\right)$, $n \in \mathbb{Z}$ works.

But because of how polar works, we could also have $r = - 9$ and $\theta = \frac{5 \pi}{3}$ and it's coterminal angles, so:

$\left(- 9 , \frac{5 \pi}{3} + 2 \pi n\right)$, $n \in \mathbb{Z}$ works as well.

We can actually combine these if we want:

$\left({\left(- 1\right)}^{n} \cdot 9 , \frac{2 \pi}{3} + \pi \cdot n\right)$, $n \in \mathbb{Z}$