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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 39
Chapter 5, Problem 39

In Exercises 35–44, test for symmetry and then graph each polar equation. r = 1 / 1−cos θ

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Identify the given polar equation: \(r = \frac{1}{1 - \cos \theta}\).
Recall the tests for symmetry in polar coordinates: - Symmetry about the polar axis (x-axis): Replace \(\theta\) by \(-\theta\) and check if the equation remains unchanged. - Symmetry about the line \(\theta = \frac{\pi}{2}\) (y-axis): Replace \(\theta\) by \(\pi - \theta\) and check if the equation remains unchanged. - Symmetry about the pole (origin): Replace \(r\) by \(-r\) and \(\theta\) by \(\theta + \pi\) and check if the equation remains unchanged.
Test for symmetry about the polar axis by substituting \(\theta\) with \(-\theta\) in the equation: \(r = \frac{1}{1 - \cos(-\theta)}\). Use the identity \(\cos(-\theta) = \cos \theta\) to simplify and compare with the original equation.
Test for symmetry about the line \(\theta = \frac{\pi}{2}\) by substituting \(\theta\) with \(\pi - \theta\): \(r = \frac{1}{1 - \cos(\pi - \theta)}\). Use the identity \(\cos(\pi - \theta) = -\cos \theta\) to simplify and compare with the original equation.
Test for symmetry about the pole by substituting \(r\) with \(-r\) and \(\theta\) with \(\theta + \pi\): \(-r = \frac{1}{1 - \cos(\theta + \pi)}\). Use the identity \(\cos(\theta + \pi) = -\cos \theta\) to simplify and compare with the original equation.

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Key Concepts

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

Polar Coordinates and Polar Equations

Polar coordinates represent points using a radius (r) and an angle (θ) from the positive x-axis. Polar equations express relationships between r and θ, describing curves in the plane. Understanding how to interpret and plot these equations is essential for graphing.
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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). These tests involve substituting θ with -θ, π - θ, or replacing r with -r to check if the equation remains unchanged, helping to predict the shape and simplify graphing.
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Cardioids

Handling and Simplifying Rational Polar Equations

Polar equations like r = 1 / (1 - cos θ) often represent conic sections and require careful algebraic manipulation to analyze. Recognizing singularities (where denominator is zero) and understanding the behavior near these points aids in accurate graphing and interpretation.
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Introduction to Common Polar Equations
Related Practice
Textbook Question

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