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Ch. 11 - Parametric Equations and Polar Coordinates
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook
All textbooksHass 15th EditionCh. 11 - Parametric Equations and Polar CoordinatesProblem 11.3.56
Chapter 11, Problem 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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Key Concepts

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

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
Related Practice
Textbook Question

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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Textbook Question

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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Textbook Question

Examples of Polar Equations


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


r = −2 cos θ

Textbook Question

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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Textbook Question

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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Textbook Question

Finding Parametric Equations


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


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