Chapter 9.5, Problem 82E

To determine

True or false:

When a same point is represented in polar coordinates system by $\left(r_1,{\theta }_1\right)$ and $\left(r_2,{\theta }_2\right)$ then ${\theta }_1={\theta }_2+2\pi$ for some integer $n$

Answer to Problem 82E

True. When a same point is represented in polar coordinates system by $\left(r_1,{\theta }_1\right)$ and $\left(r_2,{\theta }_2\right)$ then ${\theta }_1={\theta }_2+2\pi$ for some integer $n$

Explanation of Solution

Given information:

The statement is:

‘When a same point is represented in polar coordinates system by $\left(r_1,{\theta }_1\right)$ and $\left(r_2,{\theta }_2\right)$ then $\left|r_1\right|=\left|r_2\right|$ ’

It is known that, polar coordinate $\left(r,\theta \right)$ is represented for any integer $n$

  $\left(r,\theta \right)=\left(r,\theta \pm 2n\pi \right)$

Another representation for a polar point for any integer $n$ since $r$ is directed distance:

  $\left(r,\theta \right)=\left(-r,\theta \pm \left(2n+1\right)\pi \right)$

As per the given statement condition same point is represented in polar coordinates system by $\left(r_1,{\theta }_1\right)$ and $\left(r_2,{\theta }_2\right)$ then:

  $\begin{array}{l}{r}_1=r_2\\ {\theta }_1={\theta }_2\pm 2n\pi \end{array}$

Or

  $\begin{array}{l}{r}_1=-r_2\\ {\theta }_1={\theta }_2\pm \left(2n+1\right)\pi \end{array}$

So, these relations are true for any integers $n$ . Hence, the given statement is true.