Q. 19 - Calculus

Question

In Exercises 17-23 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.

$\left(5,\frac{\pi }{2}\right),\left(-5,-\frac{3\pi }{2}\right),\left(5,\frac{5\pi }{2}\right)\text{ and }\left(-5,-\frac{\pi }{2}\right)$

Step-by-Step Solution

Answer

The answer is $(0,5),(0,-5),(0,5),(0,5)$.

The graphical representation is

Step 1: Given information

Consider the polar coordinates $\left(5,\frac{\pi }{2}\right),\left(-5,-\frac{3\pi }{2}\right),\left(5,\frac{5\pi }{2}\right)\text{ and }\left(-5,-\frac{\pi }{2}\right)$.

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

Step 2: Calculation

Take $x=r\cos \theta$ and substitute $r=5,\theta =\frac{\pi }{2}$ then,

$$\begin{gathered} x=5·\cos \frac{\pi }{2} \\ x=5·0\left[since\cos \frac{\pi }{2}=0\right] \end{gathered}$$

Then the value is $x=0$.

Now take $y=r\sin \theta$ and substitute $r=5,\theta =\frac{\pi }{2}$ then,

$$\begin{gathered} y=5·\sin \frac{\pi }{2} \\ y=5·1\left[since\sin \frac{\pi }{2}=1\right] \\ y=5 \end{gathered}$$

The rectangular coordinate $(x, y)=(0,5)$.

Therefore, the rectangular coordinates are $(x, y)=(0,5)$.

For the point $\left(-5,-\frac{3\pi }{2}\right)$ then $r=-5,\theta =-\frac{3\pi }{2}$.

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

Take $x=r\cos \theta$ and substitute $r=-5,\theta =-\frac{3\pi }{2}$ then

$x=-5\cos \left(-\frac{3\pi }{2}\right)$

$x=-5·0\left[since\cos \frac{3\pi }{2}=0\right]$

Then the value is $x=0$.

Now take $y=r\sin \theta$ and substitute $r=-5,\theta =-\frac{3\pi }{2}$ then,

$$\begin{gathered} y=-5·\sin \left(-\frac{3\pi }{2}\right) \\ y=-5·1\left[since\sin \left(-\frac{3\pi }{2}\right)=1\right] \end{gathered}$$

Thus, $y=-5$.

The rectangular coordinate $(x, y)=(0,-5)$.

Therefore, the rectangular coordinates are $(x, y)=(0,-5)$.

For the point $\left(5,\frac{5\pi }{2}\right)$ then $r=5,\theta =\frac{5\pi }{2}$.

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

Take $x=r\cos \theta$ and substitute $r=5,\theta =\frac{5\pi }{2}$ then

$$\begin{gathered} x=5·\cos \frac{5\pi }{2} \\ x=5·0[\text{ since }\cos 0=1] \end{gathered}$$

Then the value is $x=0$.

Now take $y=r\sin \theta$ and substitute $r=5,\theta =\frac{5\pi }{2}$ then

$y=5·\sin \frac{5\pi }{2}$

$y=5\times 1\left[Since \sin \frac{5\pi }{2}=1\right]$

Thus, $y=5$

The rectangular coordinate $(x, y)=(0,5)$.

Therefore, the rectangular coordinates are $(x, y)=(0,5)$.

For the point $\left(-5,-\frac{\pi }{2}\right)$ then $r=-5,\theta =-\frac{\pi }{2}$.

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

Take $x=r\cos \theta$ and substitute $r=-5,\theta =-\frac{\pi }{2}$

Then

$$\begin{gathered} x=-5\cos \left(-\frac{\pi }{2}\right) \\ x=-5·0 \end{gathered}$$

Then the value is $x=0$.

Now take $y=r\sin \theta$ and substitute $r=-5,\theta =-\frac{\pi }{2}$ then

$$\begin{gathered} y=-5\sin \left(-\frac{\pi }{2}\right) \\ y=-5·-1\left[\text{ since }\sin \left(-\frac{\pi }{2}\right)=-1\right] \end{gathered}$$

Thus.

$$\begin{gathered} y=-5·-1 \\ y=5 \end{gathered}$$

The rectangular coordinate $(x, y)=(0,5)$.

Thus, the rectangular coordinates are $(x, y)=(0,5)$.

Therefore, the answer is $(0,5),(0,-5),(0,5),(0,5)$

The graphical representation is as follows