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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 12.2.43
Capitolo 12, Problema 12.2.43

37–48. Polar-to-Cartesian coordinates Convert the following equations to Cartesian coordinates. Describe the resulting curve.


r = 6 cos θ + 8 sin θ

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Recall the relationships between polar and Cartesian coordinates: \(x = r \cos \theta\) and \(y = r \sin \theta\). Also, \(r^2 = x^2 + y^2\).
Start with the given polar equation: \(r = 6 \cos \theta + 8 \sin \theta\).
Multiply both sides of the equation by \(r\) to eliminate the denominator when substituting: \(r \cdot r = r (6 \cos \theta + 8 \sin \theta)\), which gives \(r^2 = 6r \cos \theta + 8r \sin \theta\).
Substitute \(r^2\) with \(x^2 + y^2\), \(r \cos \theta\) with \(x\), and \(r \sin \theta\) with \(y\) to get: \(x^2 + y^2 = 6x + 8y\).
Rewrite the equation to standard form by bringing all terms to one side: \(x^2 - 6x + y^2 - 8y = 0\). Then, complete the square for both \(x\) and \(y\) terms to identify the type and properties of the curve.

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Polar and Cartesian Coordinate Systems

Polar coordinates represent points using a radius and an angle (r, θ), while Cartesian coordinates use (x, y) positions on a plane. Understanding how to convert between these systems is essential for analyzing curves defined in polar form.
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Intro to Polar Coordinates

Conversion Formulas Between Polar and Cartesian Coordinates

The key formulas for conversion are x = r cos θ and y = r sin θ. Additionally, r² = x² + y² and tan θ = y/x. These relationships allow rewriting polar equations in terms of x and y to identify the curve in Cartesian form.
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Intro to Polar Coordinates

Identifying and Describing Curves from Equations

After converting to Cartesian form, recognizing the type of curve (e.g., circle, ellipse, line) involves rearranging the equation and comparing it to standard forms. This helps in describing the geometric shape represented by the original polar equation.
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Convert Equations from Polar to Rectangular
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Without calculating derivatives, determine the slopes of each of the lines tangent to the curve r=8 cos θ−4 at the origin.

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39–50. Equations of ellipses and hyperbolas Find an equation of the following ellipses and hyperbolas, assuming the center is at the origin. 

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What is the polar equation of the horizontal line y = 5?

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15–22. Sets in polar coordinates Sketch the following sets of points.


2 ≤ r ≤ 8

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31–36. Eliminating the parameter Eliminate the parameter to express the following parametric equations as a single equation in x and y.


x=2 sin 8t, y=2 cos 8t 

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