Question

In Problems 31–38, plot each point given in polar coordinates $\left(-2,-\frac{2\mathrm{\pi }}{3}\right)$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 $r>0$the point is $\left(2,-\frac{5\mathrm{\pi }}{3}\right).$

Part b. When $r<0,$the point is $\left(-2,\frac{\mathrm{\pi }}{3}\right).$

Part c. When$r>0,2\mathrm{\pi }\le \mathrm{\theta }<4\mathrm{\pi }$the point is$\left(2,\frac{7\mathrm{\pi }}{3}\right).$

Step-by-step solution

Step 1. Plotting the given points in polar coordinates 

The point is $\left(-2,-\frac{2\mathrm{\pi }}{3}\right).$So, $r=-2and\theta =-\frac{2\mathrm{\pi }}{3}.$ As we can see r is negative, the ray will be in the opposite direction of$-\frac{2\mathrm{\pi }}{3}.$

The point on the polar coordinate is

Part (a) Step 1. Finding the polar coordinates 

It is given that $r>0,-2\mathrm{\pi }\le \mathrm{\theta }<0,$it means rhas to bepositive. We have to subtract $\mathrm{\pi }$from the angle $-\frac{2\mathrm{\pi }}{3}$to get the polar coordinates.

So,$\left(2,-\frac{2\mathrm{\pi }}{3}-\mathrm{\pi }\right)=\left(2,-\frac{5\mathrm{\pi }}{3}\right).$

Part (b) Step 1. Finding the polar coordinates 

It is given that $r<0,0\le \theta <2\mathrm{\pi }$, so add $\mathrm{\pi }$to angle$-\frac{2\mathrm{\pi }}{3}$to get the point.

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

Part (c) Step 1. Finding the polar coordinates 

It is given that $r>0,2\mathrm{\pi }\le \mathrm{\theta }<4\mathrm{\pi }.$ So, rhas to be positive and add $3\mathrm{\pi }$to the angle $-\frac{2\mathrm{\pi }}{3}$to get the point.

So.$\left(2,-\frac{2\mathrm{\pi }}{3}+3\mathrm{\pi }\right)=\left(2,\frac{7\mathrm{\pi }}{3}\right).$