In Exercises 59–74, convert each polar equation to a rectangular equation. Then use a rectangular coordinate system to graph the rectangular equation. r = 4 csc θ

Verified step by step guidance

1. Recall the relationship between polar and rectangular coordinates: \(x = r \cos \theta\) and \(y = r \sin \theta\). Also, \(r = \sqrt{x^2 + y^2}\).
2. Given the polar equation \(r = 4 \csc \theta\), rewrite \(\csc \theta\) in terms of sine: \(\csc \theta = \frac{1}{\sin \theta}\), so the equation becomes \(r = \frac{4}{\sin \theta}\).
3. Multiply both sides of the equation by \(\sin \theta\) to get \(r \sin \theta = 4\).
4. Substitute \(r \sin \theta\) with \(y\) (from the coordinate relationships), resulting in the rectangular equation \(y = 4\).
5. Interpret the rectangular equation \(y = 4\) as a horizontal line crossing the y-axis at 4, which can be graphed on the rectangular coordinate system.

Key Concepts

Polar and Rectangular Coordinate Systems

Polar coordinates represent points using a radius and an angle (r, θ), while rectangular coordinates use (x, y) positions on a plane. Understanding how these systems relate is essential for converting equations and graphing accurately.

Conversion Formulas Between Polar and Rectangular Coordinates

Key formulas include x = r cos θ, y = r sin θ, r² = x² + y², and tan θ = y/x. These allow transformation of polar equations into rectangular form by substituting r and θ with expressions involving x and y.

Trigonometric Functions and Their Reciprocal Identities

The cosecant function, csc θ, is the reciprocal of sin θ (csc θ = 1/sin θ). Recognizing this helps rewrite the given polar equation r = 4 csc θ in terms of sine, facilitating substitution and conversion to rectangular coordinates.