Chapter 8, Problem 2RE

(a)

To determine

The point $P\left(r,\theta \right)=P\left(8,-\frac{3\pi }{4}\right)$ is to be plotted.

Explanation of Solution

Given:

  $P\left(r,\theta \right)=P\left(8,-\frac{3\pi }{4}\right)$

Graph:

  

Interpretation:

From the given data point:

The polar co-ordinates $\left(r,\theta \right)$ of the point $P$ are $\left(r=8,\theta =-\frac{3\pi }{4}\right)$ . On the graph the point is plotte in the fourth quadrant as first value is positive and second one is negative in nature.

(b)

To determine

The values of rectangular co-ordinate for the given point $P$ are to be estimated.

Answer to Problem 2RE

The values of rectangular co-ordinate for the given point $P$ are $\left(x=-4\sqrt{2},y=-4\sqrt{2}\right)$

Explanation of Solution

Given:

  $P\left(8,-\frac{3\pi }{4}\right)$

Concept Used:

When a point is given with its polar co-ordinates $\left(r,\theta \right)$ , its rectangular co-ordinates are defined as $\left(x,y\right)$ :

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

Calculation:

From the given point is $P\left(8,-\frac{3\pi }{4}\right)$ understands that:

  $\begin{array}{l}r=8\\ \theta =-\frac{3\pi }{4}\end{array}$

Now, rectangular co-ordinates of $P$ are $\left(x,y\right)$ which are expressed as:

  $\begin{array}{l}x=r\cos \theta =8\cos \left(-\frac{3\pi }{4}\right)=-8\times \frac{\sqrt{2}}{2}=-4\sqrt{2}\\ y=r\sin \theta =8\sin \left(-\frac{3\pi }{4}\right)=-8\times \frac{\sqrt{2}}{2}=-4\sqrt{2}\end{array}$

Conclusion:

Hence, the rectangular co-ordinates are $P\left(x,y\right)=P\left(-4\sqrt{2},-4\sqrt{2}\right)$