Chapter 8, Problem 1RE

In Problems $1-3$, plot each point given in polar coordinates, and find its rectangular coordinates.

$\left(3,\frac{\pi }{6}\right)$

To determine

The rectangular coordinates for a polar coordinate $\left(3,\frac{\pi }{6}\right)$ and plot these polar coordinates $\left(3,\frac{\pi }{6}\right)$.

Answer to Problem 1RE

Solution:

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

The plot of polar coordinates $\left(3,\frac{\pi }{6}\right)$ is

Explanation of Solution

Given information:

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

Explanation:

By observing the polar coordinate it is clear that the point $\left(3,\frac{\pi }{6}\right)$ lies in first quadrent.

Below figure shows the plotting of polar coordinate $\left(3,\frac{\pi }{6}\right)$.

Compare the polar coordinate $\left(3,\frac{\pi }{6}\right)$ with standard form of polar coordinate $\left(r,\theta \right)$.

$\Rightarrow r=3,\theta =\frac{\pi }{6}$.

The rectangular form of polar coordinate $\left(r,\theta \right)$ is $x=r\cos \theta$ and $y=r\sin \theta$.

Substitute $r=3,\theta =\frac{\pi }{6}$ in equation of rectangular forms.

Therefore,

$x=3\cos \left(\frac{\pi }{6}\right)=3\left(\frac{\sqrt{3}}{2}\right)=\frac{3\sqrt{3}}{2}$,

$y=3\sin \left(\frac{\pi }{6}\right)=3\left(\frac{1}{2}\right)=\frac{3}{2}$.

Thus, the rectangular form is $\left(\frac{3\sqrt{3}}{2},\frac{3}{2}\right)$.