# What is the Cartesian form of (1, (23pi)/8 ) ?

The Cartesian Form of $\left(1 , \frac{23 \pi}{8}\right)$ is
$\left(\cos \left(\frac{7 \pi}{8}\right) , \sin \left(\frac{7 \pi}{8}\right)\right) = \left(- 0.9238795325 , 0.3826834324\right)$

#### Explanation:

The solution

from the given: Polar coordinates $\left(1 , \frac{23 \pi}{8}\right)$

Let $r = 1$ and $\theta = \frac{23 \pi}{8}$
$x = r \cos \theta$ and $y = r \sin \theta$

Let us solve for $x$

$x = r \cos \theta$
$x = \left(1\right) \cos \left(\frac{23 \pi}{8}\right)$
$x = \left(1\right) \cos \left(\frac{16 \pi}{8} + \frac{7 \pi}{8}\right)$
$x = \left(1\right) \cos \left(2 \pi + \frac{7 \pi}{8}\right)$
use the sum formula $\cos \left(A + B\right) = \cos A \cdot \cos B - \sin A \cdot \sin B$
$x = \left(1\right) \left[\cos \left(2 \pi\right) \cdot \cos \left(\frac{7 \pi}{8}\right) - \sin \left(2 \pi\right) \cdot \sin \left(\frac{7 \pi}{8}\right)\right]$
$x = \left(1\right) \left[1 \cdot \cos \left(\frac{7 \pi}{8}\right) - 0 \cdot \sin \left(\frac{7 \pi}{8}\right)\right]$

$x = \left(1\right) \cos \left(\frac{7 \pi}{8}\right)$

$x = \cos \left(\frac{7 \pi}{8}\right) = - 0.9238795325$

Let us solve for $y$

$y = r \sin \theta$
$y = \left(1\right) \sin \left(\frac{23 \pi}{8}\right)$
$y = \left(1\right) \sin \left(\frac{16 \pi}{8} + \frac{7 \pi}{8}\right)$
$y = \left(1\right) \sin \left(2 \pi + \frac{7 \pi}{8}\right)$
use the sum formula $\sin \left(A + B\right) = \sin A \cdot \cos B + \cos A \cdot \sin B$
$y = \left(1\right) \left[\sin \left(2 \pi\right) \cdot \cos \left(\frac{7 \pi}{8}\right) + \cos \left(2 \pi\right) \cdot \sin \left(\frac{7 \pi}{8}\right)\right]$
$y = \left(1\right) \left[0 \cdot \cos \left(\frac{7 \pi}{8}\right) + 1 \cdot \sin \left(\frac{7 \pi}{8}\right)\right]$

$x = \left(1\right) \cdot \sin \left(\frac{7 \pi}{8}\right)$

$x = \sin \left(\frac{7 \pi}{8}\right) = 0.3826834324$

God bless....I hope the explanation is useful.