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Ch.12 - Parametric and Polar Curves
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBriggs 3rd EditionCh.12 - Parametric and Polar CurvesProblema 12.2.79
Capitolo 12, Problema 12.2.79

Circles in general Show that the polar equation
r² - 2r r₀ cos(θ - θ₀) = R² - r₀²
describes a circle of radius R whose center has polar coordinates (r₀, θ₀)

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Recall the relationship between polar and Cartesian coordinates: \(x = r \cos\theta\) and \(y = r \sin\theta\). Similarly, the center of the circle has Cartesian coordinates \(x_0 = r_0 \cos\theta_0\) and \(y_0 = r_0 \sin\theta_0\).
Rewrite the given polar equation \(r^2 - 2 r r_0 \cos(\theta - \theta_0) = R^2 - r_0^2\) by expressing \(\cos(\theta - \theta_0)\) using the cosine difference identity: \(\cos(\theta - \theta_0) = \cos\theta \cos\theta_0 + \sin\theta \sin\theta_0\).
Substitute the expressions for \(\cos\theta\) and \(\sin\theta\) in terms of \(x\) and \(y\), and similarly for \(\cos\theta_0\) and \(\sin\theta_0\), to rewrite the equation entirely in terms of \(x\), \(y\), \(x_0\), and \(y_0\).
Simplify the equation to get it into the standard form of a circle: \((x - x_0)^2 + (y - y_0)^2 = R^2\). This will involve expanding and rearranging terms to isolate the squared terms of \((x - x_0)\) and \((y - y_0)\).
Conclude that since the equation matches the standard form of a circle with center \((x_0, y_0)\) and radius \(R\), the original polar equation indeed describes a circle with center at polar coordinates \((r_0, \theta_0)\) and radius \(R\).

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Polar Coordinates and Conversion to Cartesian Coordinates

Polar coordinates represent points using a radius and an angle (r, θ). To analyze curves like circles, it is often helpful to convert polar equations into Cartesian form using x = r cos θ and y = r sin θ. This conversion allows the use of familiar geometric interpretations and algebraic manipulations.
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Intro to Polar Coordinates

Equation of a Circle in Cartesian Coordinates

A circle with center (h, k) and radius R in Cartesian coordinates satisfies (x - h)² + (y - k)² = R². Recognizing this form after converting from polar coordinates confirms the geometric nature of the curve. Identifying the center and radius from the equation is key to understanding the shape described.
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Intro to Polar Coordinates

Use of the Law of Cosines in Polar Form

The given polar equation resembles the law of cosines, which relates the lengths of sides in a triangle to the cosine of an included angle. Interpreting r, r₀, and R as sides and θ - θ₀ as the angle helps to understand the geometric meaning of the equation and its relation to a circle centered at (r₀, θ₀).
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Convert Equations from Rectangular to Polar
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