# How do you convert [3, -5pi/4] into rectangular coordinates?

Aug 6, 2017

$\left(- 2.121 , 2.121\right)$

#### Explanation:

We're asked to find the rectangular coordinate of a given polar coordinate.

In order to do this, we use the equations

ul(x = rcostheta

ul(y = rsintheta

where in this case

• $r = 3$

• $\theta = \frac{- 5 \pi}{4}$

Thus, we have

x = 3cos((-5pi)/4) = color(red)(ul(-2.121

y = 3sin((-5pi)/4) = color(green)(ul(2.121

The coordinate is therefore

$\underline{\overline{| \stackrel{\text{ ")(" "(color(red)(-2.121), color(green)(2.121))" }}{|}}}$

Aug 6, 2017

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

#### Explanation:

$\text{to convert from "color(blue)"polar to rectangular}$

$\text{that is "(r,theta)to(x,y)" using}$

•color(white)(x)x=rcostheta" and " y=rsintheta"

$\text{here " r=3" and } \theta = - \frac{5 \pi}{4}$

$\Rightarrow x = 3 \times \cos \left(- \frac{5 \pi}{4}\right)$

$\textcolor{w h i t e}{\Rightarrow x} = - 3 \times \cos \left(\frac{\pi}{4}\right)$

$\textcolor{w h i t e}{\Rightarrow x} = - 3 \times \frac{1}{\sqrt{2}} = - \frac{3 \sqrt{2}}{2}$

$\Rightarrow y = 3 \times \sin \left(- \frac{5 \pi}{4}\right)$

$\textcolor{w h i t e}{\Rightarrow y} = 3 \times \sin \left(\frac{\pi}{4}\right)$

$\textcolor{w h i t e}{\Rightarrow y} = 3 \times \frac{1}{\sqrt{2}} = \frac{3 \sqrt{2}}{2}$

$\Rightarrow \left(3 , - \frac{5 \pi}{4}\right) \to \left(- \frac{3 \sqrt{2}}{2} , \frac{3 \sqrt{2}}{2}\right)$