Question

In Problems 31–38, plot each point given in polar coordinates $\left(1,\frac{\mathrm{\pi }}{2}\right)$and find other polar coordinates $\left(r,\theta \right)$of the point for which:

$$\begin{gathered} \left(a\right)r>0,-2\mathrm{\pi }\le \mathrm{\theta }<0 \\ \left(\mathrm{b}\right)\mathrm{r}<0,0\le \mathrm{\theta }<2\mathrm{\pi } \\ \left(\mathrm{c}\right)\mathrm{r}>0,2\mathrm{\pi }\le \mathrm{\theta }<4\mathrm{\pi } \end{gathered}$$

Short answer

The point in the polar coordinate is

Part a. When $r>0,-2\mathrm{\pi }\le \mathrm{\theta }<0,$the other polar coordinates $\left(r,\theta \right)$of the point is $\left(1,-\frac{3\mathrm{\pi }}{2}\right).$

Part b. When $\mathrm{r}<0,0\le \mathrm{\theta }<2\mathrm{\pi },$the other polar coordinates $\left(r,\theta \right)$of the point is $\left(-1,\frac{3\mathrm{\pi }}{2}\right).$

Part c. When $\mathrm{r}>0,2\mathrm{\pi }\le \mathrm{\theta }<4\mathrm{\pi },$the other polar coordinates $\left(r,\theta \right)$of the point is$\left(1,\frac{5\mathrm{\pi }}{2}\right).$

Step-by-step solution

Step 1. Plotting the given points in polar coordinates 

The point on the polar coordinate is

Part (a) Step 1. Finding the polar coordinates 

It is given that$r>0,-2\mathrm{\pi }\le \mathrm{\theta }<0.$ We have to subtract $2\mathrm{\pi }$from the angle $\frac{\mathrm{\pi }}{2}$to get the polar coordinates.

So,$\left(1,\frac{\mathrm{\pi }}{2}-2\mathrm{\pi }\right)=\left(1,-\frac{3\mathrm{\pi }}{2}\right).$

Part (b) Step 1. Finding the polar coordinates 

It is given that $r<0,0\le \theta <2\mathrm{\pi }$. Now, add $\mathrm{\pi }$to the angle $\frac{\mathrm{\pi }}{2}$ and replace the rwith negative rto get the polar coordinates.

So,$\left(-1,\frac{\mathrm{\pi }}{2}+\mathrm{\pi }\right)=\left(-1,\frac{3\mathrm{\pi }}{2}\right).$

Part (c) Step 1. Finding the polar coordinates 

It is given that $r>0,2\mathrm{\pi }\le \mathrm{\theta }<4\mathrm{\pi }.$So, $\theta$lies between $2\pi and4\pi$, to get the polar coordinate add $2\pi$to the angle $\frac{\mathrm{\pi }}{2}.$

So,$\left(1,\frac{\mathrm{\pi }}{2}+2\mathrm{\pi }\right)=\left(1,\frac{5\mathrm{\pi }}{2}\right).$