Q. 20 - Calculus

Question

The polar coordinates for several sets of points are given. Find the rectangular coordinates for each of the points, and then plot and label the points in the same polar coordinate system. 

$\left(0,\frac{\pi }{6}\right),\left(0,\frac{\pi }{3}\right),(0,\pi ) and (0,-\pi )$

Step-by-Step Solution

Answer

The  required graph is

Step 1: Given information

The polar coordinates$\left(0,\frac{\pi }{6}\right),\left(0,\frac{\pi }{3}\right),(0,\pi ),(0,-\pi )$

Step 2: Find the rectangular coordinates

Let us consider the polar coordinates $\left(0,\frac{\pi }{6}\right),\left(0,\frac{\pi }{3}\right),(0,\pi ),(0,-\pi )$

The objective is to convert the polar coordinates to the rectangular coordinates.

In the point $\left(0,\frac{\pi }{6}\right)$, then $r=0,\theta =\frac{\pi }{6}$.

Use the equations$x=r\cos \theta ,y=r\sin \theta$

Take $x=r\cos \theta$and substitute $r=0,\theta =\frac{\pi }{6}$ then

$x=0\times \cos \frac{\pi }{6}$

$x=0·\frac{\sqrt{3}}{2}\left[since\cos \frac{\pi }{6}=\frac{\sqrt{3}}{2}\right]$

Then the value is $x=0$.

Now take $y=r\sin \theta$ and substitute $r=0,\theta =\frac{\pi }{6}$ then

$$\begin{gathered} y=0\sin \frac{\pi }{6} \\ y=0\sin \frac{\pi }{6}\left[Since sin\frac{\pi }{6}=\frac{1}{2} \right] \\ y=0 \end{gathered}$$

The rectangular coordinate is $(x, y)=(0,0)$.

Therefore, the rectangular coordinates are$(x, y) = (0, 0)$.

Step 3: Simplification

For the point $\left(0,\frac{\pi }{3}\right)$ then$r=0,\theta =\frac{\pi }{3}$.

Use the equations $x=r\cos \theta ,y=r\sin \theta$,

Take$x=r\cos \theta$ and substitute $r=0,\theta =\frac{\pi }{3}$ then

$$\begin{gathered} x=0\times \cos \left(\frac{\pi }{3}\right) \\ x=0\times \frac{1}{2}\left[\text{ since }\cos \frac{\pi }{3}=\frac{1}{2}\right] \end{gathered}$$

Then the value is $x=0$.

Now take $y=r\sin \theta$and substitute $r=0,\theta =\frac{\pi }{3}$then

$y=0\sin \frac{\pi }{3}$

$y=0·\frac{\sqrt{3}}{2}\left[\text{ since }\sin \left(\frac{\pi }{3}\right)=\frac{\sqrt{3}}{2}\right]$

Thus, $y=0$.

The rectangular coordinate $(x, y)=(0,0)$.

Therefore, the rectangular coordinates are $(x, y)=(0,0)$.

For the point $(0,\pi )$then $r=0,\theta =\pi$.

Use the equations $x=r\cos \theta ,y=r\sin \theta$,

Take $x=r\cos \theta$ and substitute $r=0,\theta =\pi$ then

$$\begin{gathered} x=0\times \cos \pi \\ x=0\times -1[\text{ since }\cos \pi =-1] \end{gathered}$$

Then the value is $x=0$.

Now take $y=r\sin \theta$ and substitute $r=0,\theta =\pi$ then

$$\begin{gathered} y=0\times \sin \pi \\ y=0\times 0[since\sin \pi =0] \end{gathered}$$

Thus, $y=0$

The rectangular coordinate is $(x, y)=(0,0)$.

Therefore, the rectangular coordinates are $(x, y)=(0,0)$.

Step 4: Further simplification

For the point $(0,-\pi )$then $r=0,\theta =-\pi$

Use the equations $x=r\cos \theta ,y=r\sin \theta$.

Take$x=r\cos \theta$ and substitute $r=0,\theta =-\pi$

Then,

$$\begin{gathered} x=0\times \cos (-\pi ) \\ x=0\times -1[\text{ since }\cos \pi =-1] \end{gathered}$$

Then the value is $x=0$.

Now take$y=r\sin \theta$ and substitute $r=0,\theta =-\pi$ then

$$\begin{gathered} y=0\times \sin (-\pi ) \\ y=0\times 0[\text{ since }\sin \pi =0] \end{gathered}$$

Thus,

$y=0$

The rectangular coordinate is$(x, y)=(0,0)$.

Thus, the rectangular coordinates are $(x, y)=(0,0)$.

Therefore, the answer is $(0,0),(0,0),(0,0),(0,0).$

The graphical representation is as follows,