Chapter 41: Chapter 41

Problem 705

Find the conditions which the equation of a polar figure must satisfy for the figure to be (a) symmetric about the polar axis, (b) symmetric about the \(90^{\circ}\) -axis, (c) symmetric about the pole.

Short Answer

Expert verified

A polar equation \(r = f(θ)\) is symmetric about (a) the polar axis if \(r = f(-θ)\), (b) the \(90^{\circ}\)-axis if \(r = f(\frac{π}{2} - θ)\), and (c) the pole if \(-r = f(θ + π)\) or \(r = f(θ + π)\).

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TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Symmetric about the polar axis

To verify if a polar equation is symmetric about the polar axis, we must check if the equation remains unchanged when we replace \(θ\) by \(-θ\). So, given a polar equation \(r = f(θ)\), if \(r = f(-θ)\) holds true, then the polar equation is symmetric about the polar axis. (a) For example, let's check if the polar equation \(r = 1 + \cos(θ)\) is symmetric about the polar axis: Replace \(θ\) with \(-θ\) in the expression: \(r = 1 + \cos(-θ)\). Since \(\cos(-θ) = \cos(θ)\), the new expression becomes \(r = 1 + \cos(θ)\), which is the original equation. Therefore, the polar equation \(r = 1 + \cos(θ)\) is symmetric about the polar axis.

Step 2: Symmetric about the \(90^{\circ}\)-axis

To verify if a polar equation is symmetric about the \(90^{\circ}\)-axis, we must check if the equation remains unchanged when we replace \(θ\) by \(\frac{π}{2} - θ\). So, given a polar equation \(r = f(θ)\), if \(r = f(\frac{π}{2} - θ)\) holds true, then the polar equation is symmetric about the \(90^{\circ}\)-axis. (b) For example, let's check if the polar equation \(r = \sin(2θ)\) is symmetric about the \(90^{\circ}\)-axis: Replace \(θ\) with \(\frac{π}{2} - θ\) in the expression: \(r = \sin\left(2\left(\frac{π}{2} - θ\right)\right)\). Using the double angle identity, we get \(r = \sin(π - 4θ)\). Since \(\sin(π - 4θ) = \sin(4θ)\), the new expression becomes \(r = \sin(4θ)\), which is the original equation. Therefore, the polar equation \(r = \sin(4θ)\) is symmetric about the \(90^{\circ}\)-axis.

Step 3: Symmetric about the pole

To verify if a polar equation is symmetric about the pole, we must check if the equation remains unchanged when we replace \(r\) by \(-r\), and \(θ\) by \(θ + π\). So, given a polar equation \(r = f(θ)\), if \(-r = f(θ + π)\) or \(r = f(θ + π)\) holds true, then the polar equation is symmetric about the pole. (c) For example, let's check if the polar equation \(r = 2\cos(θ)\) is symmetric about the pole: Replace \(θ\) with \(θ + π\) in the expression: \(r = 2\cos(θ + π)\). Using the cosine addition formula, we get \(r = 2(-\cosθ) = -2\cos(θ)\). The equation \(r =-2\cos(θ)\) is equivalent to \(-r = 2\cos(θ)\). Therefore, the polar equation \(r = 2\cos(θ)\) is indeed symmetric about the pole.

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Chapter 41: Chapter 41

Find the conditions which the equation of a polar figure must satisfy for the figure to be (a) symmetric about the polar axis, (b) symmetric about the \(90^{\circ}\) -axis, (c) symmetric about the pole.

Short Answer

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Step 1: Symmetric about the polar axis

Step 2: Symmetric about the \(90^{\circ}\)-axis

Step 3: Symmetric about the pole

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