Chapter 9, Problem 3RE

To determine

To find:The rectangular coordinates and plot the graph for polar coordinated.

Answer to Problem 3RE

The rectangular coordinates for polar coordinates $\left(-3,-\frac{\pi }{2}\right)$ is $\left(0,3\right)$ .

Explanation of Solution

Given information:

The given polar coordinates are $\left(-3,-\frac{\pi }{2}\right)$ .

Calculation:

Consider a point $P$ , with polar coordinates $\left(r,\theta \right)$ , first draw angle $\theta$ , with vertex at pole and initial side along polar axis.

Here, $r=-3$ and $\theta =-\frac{\pi }{2}$ .

First draw angle $\theta =-\frac{\pi }{2}$ .

  

Figure (1)

  $r$ is negative, the ray happens to be in the opposite direction of the terminal side of $\theta$ . So, move $3$ units opposite terminal side of $-\frac{\pi }{2}$ to get point $P$ having polar coordinates $\left(-3,-\frac{\pi }{2}\right)$ as shown in figure (2).

  

Figure (2)

The rectangular coordinated for the point $P$ with polar coordinates $\left(r,\theta \right)$ are give as.

  $x=r\cos \theta$ and $y=r\sin \theta$ .

Substitute $-3$ for $r$ and $-\frac{\pi }{2}$ for $\theta$ in $x=r\cos \theta$ .

  $\begin{array}{c}x=r\cos \theta \\ =3\cos \left(-\frac{\pi }{2}\right)\\ =3\left(0\right)\\ =0\end{array}$

Substitute $-3$ for $r$ and $-\frac{\pi }{2}$ for $\theta$ in $y=r\sin \theta$ .

  $\begin{array}{c}y=r\sin \theta \\ =-3\sin \left(-\frac{\pi }{2}\right)\\ =-3\left(-\sin \frac{\pi }{2}\right)\\ =-3\left(-1\right)=3\end{array}$

Therefore, the rectangular coordinates for polar coordinates $\left(-3,-\frac{\pi }{2}\right)$ is $\left(0,3\right)$ .