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16. Parametric Equations & Polar Coordinates
Polar Coordinates
3:01 minutesProblem 12.2.52
Textbook Question
49–52. Cartesian-to-polar coordinates Convert the following equations to polar coordinates.
(x - 1)² + y² = 1
Verified step by step guidance1
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 \((x - 1)^2 + y^2 = 1\) to rewrite it in terms of \(r\) and \(\theta\).
Expand the expression: \((r \cos{\theta} - 1)^2 + (r \sin{\theta})^2 = 1\).
Simplify the equation by expanding the square and combining like terms: \((r^2 \cos^2{\theta} - 2r \cos{\theta} + 1) + r^2 \sin^2{\theta} = 1\).
Use the Pythagorean identity \(\cos^2{\theta} + \sin^2{\theta} = 1\) to combine terms and then isolate \(r\) to express the equation purely in terms of \(r\) and \(\theta\).
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3mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Polar Coordinates System
Polar coordinates represent points in the plane using a radius and an angle, denoted as (r, θ), where r is the distance from the origin and θ is the angle from the positive x-axis. This system is useful for describing curves that are circular or have rotational symmetry.
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Intro to Polar Coordinates
Conversion Formulas Between Cartesian and Polar Coordinates
To convert from Cartesian (x, y) to polar (r, θ), use x = r cos θ and y = r sin θ. Conversely, r = √(x² + y²) and θ = arctan(y/x). These formulas allow rewriting equations from Cartesian form into polar form.
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Equation of a Circle in Polar Coordinates
A circle centered at (a, 0) with radius R in Cartesian coordinates can be expressed in polar form by substituting x = r cos θ and y = r sin θ into the circle's equation. This often results in an equation involving r and θ that describes the same circle in polar terms.
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