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Ch. 5 - Complex Numbers, Polar Coordinates and Parametric Equations
Blitzer - Trigonometry 3rd Edition
Blitzer3rd EditionTrigonometryISBN: 9780137316601Not the one you use?Change textbook
All textbooksBlitzer 3rd EditionCh. 5 - Complex Numbers, Polar Coordinates and Parametric EquationsProblem 13
Chapter 5, Problem 13

Test for symmetry and then graph each polar equation. r = 2 cos θ

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1
Recall the three common tests for symmetry in polar equations: symmetry about the polar axis (x-axis), symmetry about the line \( \theta = \frac{\pi}{2} \) (y-axis), and symmetry about the pole (origin).
To test for symmetry about the polar axis, replace \( \theta \) with \( -\theta \) in the equation and check if the equation remains unchanged. For \( r = 2 \cos \theta \), substitute \( -\theta \) to get \( r = 2 \cos(-\theta) \).
Use the identity \( \cos(-\theta) = \cos \theta \) to simplify the expression. Since the equation remains the same, the graph is symmetric about the polar axis.
To test for symmetry about the line \( \theta = \frac{\pi}{2} \), replace \( \theta \) with \( \pi - \theta \) and check if the equation remains unchanged. Substitute \( \pi - \theta \) into \( r = 2 \cos \theta \) to get \( r = 2 \cos(\pi - \theta) \).
Use the identity \( \cos(\pi - \theta) = -\cos \theta \) to simplify. Since the equation changes sign, the graph is not symmetric about the line \( \theta = \frac{\pi}{2} \). Finally, test for symmetry about the pole by replacing \( r \) with \( -r \) and \( \theta \) with \( \theta + \pi \), and check if the equation remains unchanged.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Polar Coordinates and Equations

Polar coordinates represent points using a radius r and an angle θ from the positive x-axis. Polar equations express r as a function of θ, describing curves in the plane. Understanding how to interpret and plot these equations is essential for graphing polar curves.
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Intro to Polar Coordinates

Symmetry Tests in Polar Graphs

Symmetry in polar graphs can be tested about the polar axis, the line θ = π/2, and the pole (origin). For example, replacing θ with -θ tests symmetry about the polar axis, while replacing r with -r tests symmetry about the pole. Identifying symmetry simplifies graphing and analysis.
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Cardioids

Graphing Polar Equations Involving Cosine

Equations like r = 2 cos θ often represent circles or limacons. The cosine function affects the shape and orientation of the graph, typically producing curves symmetric about the polar axis. Recognizing these patterns helps in sketching accurate graphs.
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Introduction to Common Polar Equations
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