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

r = 3 csc θ

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Recall the relationships between polar and Cartesian coordinates: \(x = r \cos \theta\), \(y = r \sin \theta\), and \(r^2 = x^2 + y^2\).

Given the equation \(r = 3 \csc \theta\), rewrite \(\csc \theta\) in terms of sine: \(\csc \theta = \frac{1}{\sin \theta}\), so the equation becomes \(r = \frac{3}{\sin \theta}\).

Multiply both sides of the equation by \(\sin \theta\) to get \(r \sin \theta = 3\).

Use the polar-to-Cartesian conversion \(r \sin \theta = y\) to rewrite the equation as \(y = 3\).

Interpret the Cartesian equation \(y = 3\): this represents a horizontal line located 3 units above the x-axis.

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Polar and Cartesian Coordinate Systems

Polar coordinates represent points using a radius and an angle (r, θ), while Cartesian coordinates use x and y values. Understanding how these systems relate is essential for converting equations between them, as each system describes points in the plane differently.

Conversion Formulas Between Polar and Cartesian Coordinates

The key formulas for conversion are x = r cos θ and y = r sin θ. These allow expressing polar equations in terms of x and y by substituting r and θ, enabling the transformation of polar curves into Cartesian form for easier analysis.

Trigonometric Identities and Curve Description

Using identities like csc θ = 1/sin θ helps rewrite the given equation. After conversion, recognizing the resulting Cartesian equation (e.g., a line, circle, or parabola) is crucial to describe the curve's shape and properties accurately.

37–48. Polar-to-Cartesian coordinates Convert the following equations to Cartesian coordinates. Describe the resulting curve.r = 3 csc θ