# How do you find the rectangular coordinates of the point with polar coordinates (-1,pi/3)?

Sep 27, 2014

When trying to convert polar coords to rectangular coords or vice versa you need to be aware of a few equations.

$\cos \left(\theta\right) = \frac{x}{r} \to x = r \cos \left(\theta\right)$

$\sin \left(\theta\right) = \frac{y}{r} \to y = r \sin \left(\theta\right)$

$\tan \left(\theta\right) = \frac{y}{x} \to \theta = {\tan}^{-} 1 \left(\frac{y}{x}\right)$

${x}^{2} + {y}^{2} = {r}^{2} \to r = \sqrt{{x}^{2} + {y}^{2}}$

Polar Coordinates

$\left(r , \theta\right) \to \left(- 1 , \frac{\pi}{3}\right)$

Calculate the $x$-value

$x = r \cos \left(\theta\right)$

$x = \left(- 1\right) \cdot \cos \left(\frac{\pi}{3}\right)$

$x = \left(- 1\right) \left(\frac{1}{2}\right)$

$x = - \frac{1}{2}$

Calculate the $y$-value

$y = r \sin \left(\theta\right)$

$y = \left(- 1\right) \left(\frac{\sqrt{3}}{2}\right)$

$y = - \frac{\sqrt{3}}{2}$

Rectangular Coordinates

$\left(x , y\right) \to \left(- \frac{1}{2} , - \frac{\sqrt{3}}{2}\right)$