What is the Cartesian form of ( -1, (4pi)/3 ) ?

Jul 18, 2018

$\left(\frac{1}{2} , \frac{\sqrt{3}}{2}\right)$

Explanation:

We are given the polar form, so there is a radius and an angle. We want to convert to $x , y$ coordinates.

So we can use Euler's formula (or at least the idea behind it) to convert between Cartesian and polar:

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

From that, we just plug in the numbers, remembering our unit circle:

$\cos \left(\frac{4 \pi}{3}\right) = - \frac{1}{2} \mathmr{and} \sin \left(\frac{4 \pi}{3}\right) = - \frac{\sqrt{3}}{2}$
therefore
$\left(x , y\right) = \left(\frac{1}{2} , \frac{\sqrt{3}}{2}\right)$
You could also notice that a negative radius is the same as adding or subtracting $\pi$ to the angle, hence
$\left(- 1 , \frac{4 \pi}{3}\right) = \left(1 , \frac{\pi}{3}\right)$
which I think is a bit easier to think about.

$\left(\frac{1}{2} , \setminus \frac{\sqrt{3}}{2}\right)$

Explanation:

The Cartesian coordinates $\left(x , y\right)$ of the point $\left(- 1 , \frac{4 \setminus \pi}{3}\right) \setminus \equiv \left(r , \setminus \theta\right)$ are given as follows

$x = r \setminus \cos \setminus \theta$

$= - 1 \setminus \cos \left(\frac{4 \setminus \pi}{3}\right)$

$= - \setminus \cos \left(\setminus \pi + \setminus \frac{\pi}{3}\right)$

$= \setminus \cos \left(\setminus \frac{\pi}{3}\right)$

$= \frac{1}{2}$

$y = r \setminus \sin \setminus \theta$

$= - 1 \setminus \sin \left(\frac{4 \setminus \pi}{3}\right)$

$= - \setminus \sin \left(\setminus \pi + \setminus \frac{\pi}{3}\right)$

$= \setminus \sin \left(\setminus \frac{\pi}{3}\right)$

$= \setminus \frac{\sqrt{3}}{2}$

hence, the Cartesian coordinates are $\left(\frac{1}{2} , \setminus \frac{\sqrt{3}}{2}\right)$