# How do you convert (-2, -2sqrt{3})into polar coordinates?

May 5, 2018

$\left(4 , \frac{4 \pi}{3}\right)$ (radians) or $\left(4 , {240}^{\circ}\right)$ (degrees)

#### Explanation:

Rectangular $\to$ Polar: $\left(x , y\right) \to \left(r , \theta\right)$

• Find $r$ (radius) using $r = \sqrt{{x}^{2} + {y}^{2}}$
• Find $\theta$ by finding the reference angle: $\tan \theta = \frac{y}{x}$ and use this to find the angle in the correct quadrant

$r = \sqrt{{\left(- 2\right)}^{2} + {\left(- 2 \sqrt{3}\right)}^{2}}$

$r = \sqrt{4 + \left(4 \cdot 3\right)}$

$r = \sqrt{4 + 12}$

$r = \sqrt{16}$

$r = 4$

Now we find the value of $\theta$ using $\tan \theta = \frac{y}{x}$.

$\tan \theta = \frac{- 2 \sqrt{3}}{-} 2$

$\tan \theta = \sqrt{3}$

$\theta = {\tan}^{-} 1 \left(\sqrt{3}\right)$

$\theta = \frac{\pi}{3}$ or $\frac{4 \pi}{3}$

To determine which one it is, we have to look at our coordinate $\left(- 2 , - 2 \sqrt{3}\right)$. First, let's graph it:

As you can see, it is in the third quadrant. Our $\theta$ has to match that quadrant, meaning that $\theta = \frac{4 \pi}{3}$.

From $r$ and $\theta$, we can write our polar coordinate:
$\left(4 , \frac{4 \pi}{3}\right)$ (radians) or $\left(4 , {240}^{\circ}\right)$ (degrees)

Hope this helps!