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Ch.12 - Parametric and Polar Curves
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh.12 - Parametric and Polar CurvesProblem 12.2.50
Chapter 12, Problem 12.2.50

49–52. Cartesian-to-polar coordinates Convert the following equations to polar coordinates.


y = 3

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Recall the relationships between Cartesian coordinates \((x, y)\) and polar coordinates \((r, \theta)\): \(x = r \cos{\theta}\) and \(y = r \sin{\theta}\).
Given the equation \(y = 3\), substitute \(y\) with its polar form: \(r \sin{\theta} = 3\).
Isolate \(r\) to express it in terms of \(\theta\): \(r = \frac{3}{\sin{\theta}}\).
Note the domain restrictions where \(\sin{\theta} \neq 0\) to avoid division by zero.
The polar form of the equation \(y = 3\) is therefore \(r = \frac{3}{\sin{\theta}}\).

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Key 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 measured from the positive x-axis. This system is useful for describing curves that are difficult to express in Cartesian coordinates.
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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 r = √(x² + y²) and θ = arctan(y/x). Conversely, x = r cos(θ) and y = r sin(θ). These formulas allow rewriting equations from one coordinate system to the other.
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Intro to Polar Coordinates

Expressing Cartesian Equations in Polar Form

To convert a Cartesian equation like y = 3 into polar form, substitute y with r sin(θ). This transforms the equation into r sin(θ) = 3, which can then be analyzed or manipulated using polar coordinates.
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Introduction to Common Polar Equations
Related Practice
Textbook Question

31–36. Converting coordinates Express the following Cartesian coordinates in polar coordinates in at least two different ways.


(-4, 4√3)

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Textbook Question

Subtle symmetry Without using a graphing utility, determine the symmetries (if any) of the curve r=4-sin (θ/2)

Textbook Question

81–88. Arc length Find the arc length of the following curves on the given interval.


x = 3 cos t, y = 3 sin t + 1; 0 ≤ t ≤ 2π

Textbook Question

93–94. Parametric equations of ellipses Find parametric equations (not unique) of the following ellipses (see Exercises 91–92). Graph the ellipse and find a description in terms of x and y.


An ellipse centered at (-2, -3) with major and minor axes of lengths 30 and 20, parallel to the x- and y-axes, respectively, generated counterclockwise (Hint: Shift the parametric equations.)

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Textbook Question

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


r = 6 cos θ + 8 sin θ

Textbook Question

13–30. Graphing conic sections Determine whether the following equations describe a parabola, an ellipse, or a hyperbola, and then sketch a graph of the curve. For each parabola, specify the location of the focus and the equation of the directrix; for each ellipse, label the coordinates of the vertices and foci, and find the lengths of the major and minor axes; for each hyperbola, label the coordinates of the vertices and foci, and find the equations of the asymptotes.


x² + y²/9 = 1