Textbook Question
49–52. Cartesian-to-polar coordinates Convert the following equations to polar coordinates.
y = 3
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16. Parametric Equations & Polar Coordinates
Polar Coordinates
2:21 minutesProblem 12.2.37
Textbook Question
37–48. Polar-to-Cartesian coordinates Convert the following equations to Cartesian coordinates. Describe the resulting curve.
r cos θ = -4
Verified step by step guidance1
Recall the relationships between polar and Cartesian coordinates: \(x = r \cos \theta\) and \(y = r \sin \theta\).
Given the equation \(r \cos \theta = -4\), substitute \(x\) for \(r \cos \theta\) to rewrite the equation in Cartesian form.
After substitution, the equation becomes \(x = -4\).
Recognize that \(x = -4\) represents a vertical line in the Cartesian coordinate plane where all points have an \(x\)-coordinate of \(-4\).
Therefore, the curve described by the polar equation \(r \cos \theta = -4\) is a vertical line located 4 units to the left of the \(y\)-axis.
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Video duration:
2mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Polar and Cartesian Coordinate Systems
Polar coordinates represent points using a radius and an angle (r, θ), while Cartesian coordinates use (x, y) positions on a plane. Understanding how these systems relate is essential for converting equations between them.
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Intro to Polar Coordinates
Conversion Formulas Between Polar and Cartesian Coordinates
The key formulas are x = r cos θ and y = r sin θ. These allow conversion from polar to Cartesian form by expressing r and θ in terms of x and y, enabling the rewriting of polar equations into Cartesian equations.
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Interpreting Cartesian Equations to Identify Curves
Once converted, the Cartesian equation can be analyzed to identify the type of curve it represents, such as lines, circles, or parabolas. Recognizing standard forms helps describe the geometric nature of the curve.
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04:47Introduction to Parametric Equations
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