Chapter 8, Problem 3RE

(a)

To determine

The point $P\left(-3,\frac{7\pi }{4}\right)$ is to be plotted.

Explanation of Solution

Given:

  $P\left(-3,\frac{7\pi }{4}\right)$

Graph:

  

Interpretation:

From the given data point:

The polar co-ordinates $\left(r,\theta \right)$ of the point $P$ are $\left(r=-3,\theta =\frac{7\pi }{4}\right)$ . On the graph the point is plotted in the second quadrant as first value is negative and second one is positive in nature.

(b)

To determine

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

Answer to Problem 3RE

The values of rectangular co-ordinate for the given point $P$ are $\left(x=-\frac{3}{\sqrt{2}},y=\frac{3}{\sqrt{2}}\right)$

Explanation of Solution

Given:

  $P\left(-3,\frac{7\pi }{4}\right)$

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 $P\left(-3,\frac{7\pi }{4}\right)$ understands that:

  $\begin{array}{l}r=-3\\ \theta =\frac{7\pi }{4}\end{array}$

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

  $\begin{array}{l}x=r\cos \theta =-3\cos \left(\frac{7\pi }{4}\right)=-3\times \frac{1}{\sqrt{2}}=-\frac{3}{\sqrt{2}}\\ y=r\sin \theta =-3\sin \left(\frac{7\pi }{4}\right)=3\times \frac{1}{\sqrt{2}}=\frac{3}{\sqrt{2}}\end{array}$

Conclusion:

Hence, the rectangular co-ordinates are $P\left(x=-\frac{3}{\sqrt{2}},y=\frac{3}{\sqrt{2}}\right)$