Question

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

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

Short answer

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

Step-by-step solution

Step 1. Given information

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

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

$$\begin{gathered} x=r\cos \theta \\ =-3.\cos (-\frac{3\mathrm{\pi }}{4}) \\ =-3.(-\frac{1}{\sqrt{2}}) \\ =\frac{3}{\sqrt{2}} \end{gathered}$$

Step 3. Finding the value of y.

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

$$\begin{gathered} y=r\sin \theta \\ =-3.\sin (-\frac{3\mathrm{\pi }}{4}) \\ =-3\times -\frac{1}{\sqrt{2}} \\ =\frac{3}{\sqrt{2}} \end{gathered}$$

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