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Ch. 11 - Parametric Equations and Polar Coordinates
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoHass 15th EditionCh. 11 - Parametric Equations and Polar CoordinatesProblema 11.3.56
Capitolo 11, Problema 11.3.56

Cartesian to Polar Equations


Replace the Cartesian equations in Exercises 53–66 with equivalent polar equations.


x - y = 3

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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 \(x\) and \(y\) in the given Cartesian equation \(x - y = 3\) with their polar equivalents: \(r \cos{\theta} - r \sin{\theta} = 3\).
Factor out \(r\) from the left side to get \(r (\cos{\theta} - \sin{\theta}) = 3\).
Solve for \(r\) by dividing both sides by \((\cos{\theta} - \sin{\theta})\), yielding \(r = \frac{3}{\cos{\theta} - \sin{\theta}}\).
This expression represents the equivalent polar equation for the given Cartesian equation.

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Cartesian and Polar Coordinate Systems

Cartesian coordinates represent points using (x, y) values on perpendicular axes, while polar coordinates use (r, θ), where r is the distance from the origin and θ is the angle from the positive x-axis. Understanding both systems is essential to convert equations between them.
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Intro to Polar Coordinates

Conversion Formulas Between Cartesian and Polar Coordinates

The key formulas for conversion are x = r cos(θ) and y = r sin(θ). These allow substitution of Cartesian variables with polar expressions, enabling the rewriting of Cartesian equations in terms of r and θ.
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Intro to Polar Coordinates

Algebraic Manipulation for Equation Conversion

After substituting x and y with their polar equivalents, algebraic manipulation is required to simplify and express the equation purely in terms of r and θ. This may involve factoring, isolating r, or using trigonometric identities.
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Parameterizing Equations
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Tangent Lines to Parametrized Curves


In Exercises 1−14, find an equation for the line tangent to the curve at the point defined by the given value of t. Also, find the value of d²y/dx² at this point.


x = t + eᵗ, y = 1 − eᵗ, t = 0

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Lengths of Curves


Find the lengths of the curves in Exercises 25–30.


x = cos t, y = t + sin t, 0 ≤ t ≤ π

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Examples of Polar Equations


[Technology Exercise] Graph the lines and conic sections in Exercises 65–74.


r = −2 cos θ

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Domanda del libro di testo

Polar to Cartesian Equations


Replace the polar equations in Exercises 27–52 with equivalent Cartesian equations. Then describe or identify the graph.


r² = 4r sin θ

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Finding Parametric Equations


In Exercises 31–36, find a parametrization for the curve.


the ray (half line) with initial point (-1,2) that passes through the point (0,0)

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Domanda del libro di testo

Finding Parametric Equations


In Exercises 31–36, find a parametrization for the curve.


the line segment with endpoints (-1,3) and (3,-2)

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