Ch.12 - Parametric and Polar Curves

Briggs - Calculus: Early Transcendentals 3rd Edition

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Briggs 3rd Edition

Ch.12 - Parametric and Polar Curves

Problem 12.R.38

Chapter 12, Problem 12.R.38

Cartesian conversion Write the equation x=y ² in polar coordinates and state values of θ that produce the entire graph of the parabola.

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Recall the relationships between Cartesian coordinates \((x, y)\) and polar coordinates \((r, \theta)\): \(x = r \cos{\theta}\) and \(y = r \sin{\theta}\).

Substitute \(x = r \cos{\theta}\) and \(y = r \sin{\theta}\) into the given equation \(x = y^2\) to get \(r \cos{\theta} = (r \sin{\theta})^2\).

Simplify the right side: \((r \sin{\theta})^2 = r^2 \sin^2{\theta}\), so the equation becomes \(r \cos{\theta} = r^2 \sin^2{\theta}\).

Rearrange the equation to isolate \(r\): \(r^2 \sin^2{\theta} - r \cos{\theta} = 0\), which can be factored as \(r (r \sin^2{\theta} - \cos{\theta}) = 0\).

Solve for \(r\) (ignoring the trivial solution \(r=0\)): \(r = \frac{\cos{\theta}}{\sin^2{\theta}}\). Then, analyze the values of \(\theta\) for which this expression is defined and produces the entire parabola.

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Key Concepts

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

Polar Coordinates

Polar coordinates represent points in the plane using a radius r and an angle θ, where x = r cos θ and y = r sin θ. This system is useful for converting Cartesian equations into forms involving r and θ, facilitating analysis of curves with radial symmetry.

Conversion of Cartesian to Polar Equations

To convert Cartesian equations to polar form, substitute x = r cos θ and y = r sin θ into the equation. This process transforms the equation into one involving r and θ, allowing the curve to be expressed in terms of polar variables.

Domain of θ for Complete Graph Representation

When expressing curves in polar form, the range of θ must be chosen to cover all points of the original graph. Identifying the correct interval for θ ensures the entire curve is traced without omission or repetition.

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