# What is the Cartesian form of (-10,(14pi)/8))?

Jan 14, 2018

See a solution process below:

#### Explanation:

The formula for converting Polar Coordinates to Cartesian coordinates is:

For, $\left(r , \theta\right)$; $x = r \times \cos \left(\theta\right)$; $y = r \times \sin \left(\theta\right)$

Substituting the coordinates from the problem gives:

For, $\left(- 10 , \frac{14 \pi}{8}\right)$:

$x = - 10 \times \cos \left(\frac{14 \pi}{8}\right) = - 10 \times 0.707 = - 70.7$

$y = - 10 \times \sin \left(\frac{14 \pi}{8}\right) = - 10 \times - 0.707 = 70.7$

$\left(- 10 , \frac{14 \pi}{8}\right) = \left(- 70.7 , 70.7\right)$

Jan 14, 2018

The cartesian form is $\left(- 5 \sqrt{2} , 5 \sqrt{2}\right)$.

#### Explanation:

To convert polar to rectangular we use:

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

We're given the polar point $\left(- 10 , \frac{14 \pi}{8}\right)$.

First let's simplify $\frac{14 \pi}{8}$ to $\frac{7 \pi}{4}$.

From what we have we know:

$x = \left(- 10\right) \cos \left(\frac{7 \pi}{4}\right)$
$y = \left(- 10\right) \sin \left(\frac{7 \pi}{4}\right)$

$\frac{7 \pi}{4}$ is a Unit Circle angle so we know its sine and cosine.

$\cos \left(\frac{7 \pi}{4}\right) = \frac{\sqrt{2}}{2}$
$\sin \left(\frac{7 \pi}{4}\right) = - \frac{\sqrt{2}}{2}$

so:

$x = \left(- 10\right) \left(\frac{\sqrt{2}}{2}\right) = - 5 \sqrt{2}$
$y = \left(- 10\right) \left(- \frac{\sqrt{2}}{2}\right) = 5 \sqrt{2}$

so the cartesian form is $\left(- 5 \sqrt{2} , 5 \sqrt{2}\right)$.