Chapter 9.1, Problem 31AYU

In Problems 33-40, plot each point given in polar coordinates, and find other polar coordinates $\left(r,\theta \right)$ of the point for which:

a. $r\text{ > 0, }-2\pi \le \theta \text{ < 0}$

b. $r\text{ < 0, }0\le \theta \text{ < 2}\pi$

c. $r\text{ > 0, }2\pi \le \theta \text{ < 4}\pi$

$\left(5,\frac{2\pi }{3}\right)$

To determine

To plot: Each point given in polar coordinates and find other polar coordinates $\left(r,\theta \right)$ of the point for which:

a. $r>0,-2\pi \le \theta <0$

Answer to Problem 31AYU

a.

Explanation of Solution

Given:

$\left(5,\frac{2\pi }{3}\right)$

Graph:

Calculation:

The point $\left(5,\frac{2\pi }{3}\right)$ is plotted in the graph:

a. Subtract $\left(2\pi \text{radians}\right)$ from the angle $\frac{2\pi }{3}$ to get $\text{P }=\left(5,\frac{2\pi }{3}-2\pi \right)=\left(5,\frac{-4\pi }{3}\right)$.

Graph:

a.

To determine

To plot: Each point given in polar coordinates and find other polar coordinates $\left(r,\theta \right)$ of the point for which:

b. $r<0,0\le \theta <2\pi$

Answer to Problem 31AYU

b.

Explanation of Solution

Given:

$\left(5,\frac{2\pi }{3}\right)$

Graph:

Calculation:

The point $\left(5,\frac{2\pi }{3}\right)$ is plotted in the graph:

b. Add $\frac{1}{2}$ revolution $\left(\pi \text{radians}\right)$ to the angle $\frac{2\pi }{3}$, and replace 5 by $-5$ to get $\text{P }=\left(-5,\frac{2\pi }{3}+\pi \right)=\left(-5,\frac{5\pi }{3}\right)$.

Graph:

b.

To determine

To plot: Each point given in polar coordinates and find other polar coordinates $\left(r,\theta \right)$ of the point for which:

c. $r>0,2\pi \le \theta <4\pi$

Answer to Problem 31AYU

c.

Explanation of Solution

Given:

$\left(5,\frac{2\pi }{3}\right)$

Graph:

Calculation:

The point $\left(5,\frac{2\pi }{3}\right)$ is plotted in the graph:

c. Add 1 revolution $\left(2\pi \text{radians}\right)$ to the angle $\frac{2\pi }{3}$ to get $\text{P }=\left(5,\frac{2\pi }{3}+2\pi \right)=\left(5,\frac{8\pi }{3}\right)$.

Graph:

c.