Chapter 6.4, Problem 15E

Solution

To find: The rectangular coordinates with the given polar coordinates.

The rectangular coordinate is $\left(\frac{3}{4},\frac{3\sqrt{3}}{4}\right)$ .

Given data:

The polar coordinate is $\left(1.5,\frac{7\pi }{3}\right)$ .

Formula used:

The rectangular coordinate of the polar coordinate $\left(r,\theta \right)$ is given by,

  $\begin{array}{c}x=r\cos \theta \\ y=r\sin \theta \end{array}$

Calculation:

Substitute $1.5$ for $r$, and $\frac{7\pi }{3}$ for $\theta$ in $x=r\cos \theta$ .

  $\begin{array}{c}x=1.5\cos \frac{7\pi }{3}\\ =\frac{3}{2}\left(\frac{1}{2}\right)\\ =\frac{3}{4}\end{array}$

Substitute $1.5$ for $r$, and $\frac{7\pi }{3}$ for $\theta$ in $y=r\sin \theta$ .

  $\begin{array}{c}y=1.5\sin \frac{7\pi }{3}\\ =\frac{3}{2}\left(\frac{\sqrt{3}}{2}\right)\\ =\frac{3\sqrt{3}}{4}\end{array}$

Therefore, the rectangular coordinate is $\left(\frac{3}{4},\frac{3\sqrt{3}}{4}\right)$ .