Question

In Problems 39-54,the poplar coordinates of a point are given.Find rectangular coordinates of each point.

$\left(-2,\frac{3\mathrm{\pi }}{4}\right)$

Short answer

The rectangular coordinates corresponding to the polar coordinates$(-2,\frac{3\mathrm{\pi }}{4})$are$(\sqrt{2},-\sqrt{2)}$

Step-by-step solution

Step 1. Given information

The given poplar coordinates of a point$\left(-2,\frac{3\mathrm{\pi }}{4}\right)$

Step 2. Finding the value of x.

For a point P on the terminal side of $\theta$and having polar coordinates $(r,\theta )$, the corresponding rectangular coordinates are given by $x=r\cos \theta$and$y=r\sin \theta$.
Substitute $-2$ for r and $\frac{3\mathrm{\pi }}{4}$for $\theta$in $x=r\cos \theta$ to get the value of x.

$$\begin{gathered} x=r\cos \theta \\ =-2.\cos \left(\frac{3\mathrm{\pi }}{4}\right) \\ =-2.-\frac{\sqrt{2}}{2} \\ =\sqrt{2} \end{gathered}$$

Step 3. Finding the value of y.

Substitute $-2$ for r and $\frac{3\mathrm{\pi }}{4}$for $\theta$ in $y=r\sin \theta$ to get the value of y.

$$\begin{gathered} y=r\sin \theta \\ =-2.\sin \frac{3\mathrm{\pi }}{4} \\ =-2.\frac{\sqrt{2}}{2} \\ =-\sqrt{2} \end{gathered}$$

Therefore, the rectangular coordinates corresponding to the polar coordinates $(-2,\frac{3\mathrm{\pi }}{4})$are$(\sqrt{2},-\sqrt{2)}$ .