# How do you find the polar coordinate for (-4, -4)?

May 5, 2018

$\left(4 \sqrt{2} , \frac{5 \pi}{4}\right)$ (radians) or $\left(4 \sqrt{2} , {225}^{\circ}\right)$ (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

$r = \sqrt{{\left(- 4\right)}^{2} + {\left(- 4\right)}^{2}}$

$r = \sqrt{\left(16\right) + 16}$

$r = \sqrt{32}$

$r = \sqrt{16 \cdot 2}$

$r = 4 \sqrt{2}$

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

$\tan \theta = \frac{- 4}{- 4}$

$\tan \theta = 1$

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

$\theta = \frac{\pi}{4}$ or $\frac{5 \pi}{4}$

To determine which one it is, we have to look at our coordinate $\left(- 4 , - 4\right)$. First, let's graph it:

As you can see, it is in the third quadrant. Our $\theta$ has to match that quadrant, meaning that $\theta = \frac{5 \pi}{4}$.

From $r$ and $\theta$, we can write our polar coordinate:
$\left(4 \sqrt{2} , \frac{5 \pi}{4}\right)$ (radians) or $\left(4 \sqrt{2} , {225}^{\circ}\right)$ (degrees)

Hope this helps!