Question

In Problems 31–38, plot each point given in polar coordinates $\left(-3,4\mathrm{\pi }\right)$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 on the polar coordinate is

Part a. When $r>0,-2\mathrm{\pi }\le \mathrm{\theta }<0$the polar coordinates are $\left(3,2\mathrm{\pi }\right).$

Part b. When $r>0,2\mathrm{\pi }\le \mathrm{\theta }<4\mathrm{\pi }$the polar coordinates are $\left(-3,2\mathrm{\pi }\right).$

Part c. When$r>0,2\mathrm{\pi }\le \mathrm{\theta }<4\mathrm{\pi }$the polar coordinates are$\left(3,3\mathrm{\pi }\right).$

Step-by-step solution

Step 1. Plotting the given points in polar coordinates 

The point is $\left(-3,4\mathrm{\pi }\right).$ So, $r=-3and\theta =4\pi .$As we can see ris negative, the ray will be in the opposite direction of$4\pi .$

The point on the polar coordinate is

Part (a) Step 1. Finding the polar coordinates 

It is given that $r>0,-2\mathrm{\pi }\le \mathrm{\theta }<0.$ So, r has to be positive. We have to subtract $2\mathrm{\pi }$from the angle $4\mathrm{\pi }$to get the polar coordinates.

So,$\left(3,4\mathrm{\pi }-2\mathrm{\pi }\right)=\left(3,2\mathrm{\pi }\right).$

Part (b) Step 1. Finding the polar coordinates 

It is given that $r<0,0\le \theta <2\mathrm{\pi },$now $\theta$lies between role="math" localid="1646960724964" $0and2\mathrm{\pi },$subtract$2\mathrm{\pi }$from the angle$4\mathrm{\pi }.$

So,$\left(-3,4\mathrm{\pi }-2\mathrm{\pi }\right)=\left(-3,2\mathrm{\pi }\right).$

Part (c) Step 1. Finding the polar coordinates 

It is given that $r>0,2\mathrm{\pi }\le \mathrm{\theta }<4\mathrm{\pi }$, ris greater than zero so rwill be positive and subtract $\mathrm{\pi }$from the angle $4\mathrm{\pi }$to get the polar coordinates.

So,$\left(3,4\mathrm{\pi }-\mathrm{\pi }\right)=\left(3,3\mathrm{\pi }\right).$