Q 66. - Calculus

Question

In this problem you will prove the three parts of Theorem 9.5:

(a) Prove that the polar coordinates (r, θ + 2πk) represent the same point for every integer k.

(b) Prove that the point with polar coordinates (−r, θ + π) represents the same point as (r, θ) for any value of θ.

(c) Prove that the polar coordinates (0, θ) represent the pole for any value of θ.

Step-by-Step Solution

Answer

Part (a) The representation of the point $\left(2,\frac{\pi }{6}\right)$ is same as $\left(2,\frac{13\pi }{6}\right)$

Part (b) The point $\left(2,\frac{\pi }{6}\right)$ and$\left(-2,\frac{7\pi }{6}\right)$ represents the same point.

Part (c) The points $(0,\theta )\left(0,\frac{\pi }{3}\right)$ represent the pole.

Part (a) Step 1: Given information

$(r,\theta )$

Part (a) Step 2: Concept

A polar curve is a shape constructed using the polar coordinate system.

Part (a) Step 3: Calculation

Consider the polar coordinate $(r,\theta )$

The goal is to show that for any $k$ $(r, theta+2 pi k)$ denotes the same point.

The point $(r, theta+2 pi k)$ completes the revolutions and reaches the same position for any value of the integer $k$

That is, if $k=1$ one revolution is completed. It completes two revolutions when $k=2$

Thus for any point $(r,\theta )$ the point $(r,\theta +2\pi k)$ gives the same point since we are adding $2\pi$ multiple.

In the graph, we can observe that 

Example: consider the point $\left(2,\frac{\pi }{6}\right)$

$$\begin{gathered} \left(2,\frac{\pi }{6}\right)=\left(2,\frac{\pi }{6}+2\pi \right) \\ =\left(2,\frac{13\pi }{6}\right) \end{gathered}$$

The representation of the point $\left(2,\frac{\pi }{6}\right)$ is same as $\left(2,\frac{13\pi }{6}\right)$

This is the explanation. 

Part (b) Step 1: Calculation

Consider the point $(r,\theta )$

Objective is to prove that $(-r,\theta +\pi )$ is same as $(r,\theta )$

If we add an angle $\pi$ to an angle $\theta$ in the polar coordinate system, we get the same angle.

Example: consider the polar coordinate $\left(2,\frac{\pi }{6}\right)$

$$\begin{gathered} \left(2,\frac{\pi }{6}\right)=\left(-2,\frac{\pi }{6}+\pi \right) \\ =\left(-2,\frac{7\pi }{6}\right) \end{gathered}$$

Therefore the point $\left(2,\frac{\pi }{6}\right)$ and $\left(-2,\frac{7\pi }{6}\right)$ represents the same point. This is the explanation.

Part (c) Step 1: Calculation

Consider the point $(0,\theta )$

Objective is to prove that $(0,\theta )$ is the pole for any value of $\theta$

From the graph we can observe that for any value of $\theta$ let the points $(0,\theta )\left(0,\frac{\pi }{3}\right)$ represent the pole.

This is the explanation.