Chapter 9, Problem 1RE

To determine

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

Answer to Problem 1RE

The rectangular coordinates for polar coordinates $\left(3,\frac{\pi }{6}\right)$ are $\left(\frac{3\sqrt{3}}{2},\frac{3}{2}\right)$ .

Explanation of Solution

Given information:

The given polar coordinates are $\left(3,\frac{\pi }{6}\right)$ .

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 $\theta =\frac{\pi }{6}$ .

First draw angle $\theta =\frac{\pi }{6}$ .

  

Figure (1)

Then move $3$ units along the terminal side of $\frac{\pi }{6}$ to get point $P$ having polar coordinates $\left(3,\frac{\pi }{6}\right)$ 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 $\frac{\pi }{6}$ for $\theta$ in $x=r\cos \theta$ .

  $\begin{array}{c}x=r\cos \theta \\ =3\cos \frac{\pi }{6}\\ =3\left(\frac{\sqrt{3}}{2}\right)\\ =\frac{3\sqrt{3}}{2}\end{array}$

Substitute $3$ for $r$ and $\frac{\pi }{6}$ for $\theta$ in $y=r\sin \theta$ .

  $\begin{array}{c}y=r\sin \theta \\ =3\sin \frac{\pi }{6}\\ =3\left(\frac{1}{2}\right)\\ =\frac{3}{2}\end{array}$

Therefore, the rectangular coordinates for polar coordinates $\left(3,\frac{\pi }{6}\right)$ is $\left(\frac{3\sqrt{3}}{2},\frac{3}{2}\right)$ .