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

r cos θ = -4

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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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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.

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.

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.

37–48. Polar-to-Cartesian coordinates Convert the following equations to Cartesian coordinates. Describe the resulting curve.r cos θ = -4