Ch. 5 - Complex Numbers, Polar Coordinates and Parametric Equations

Blitzer3rd EditionTrigonometryISBN: 9780137316601Non è quello che usi tu?Cambia libro di testo

Blitzer 3rd Edition

Ch. 5 - Complex Numbers, Polar Coordinates and Parametric Equations

Problema 39

Capitolo 5, Problema 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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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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Percorso guidato

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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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Percorso guidato

3:47

Introduction to Common Polar Equations

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