In Exercises 54–60, convert each polar equation to a rectangular equation. Then use your knowledge of the rectangular equation to graph the polar equation in a polar coordinate system. r = 5 csc θ

Verified step by step guidance

1. Recall the relationships between polar and rectangular coordinates: \(x = r \cos \theta\), \(y = r \sin \theta\), and \(r^2 = x^2 + y^2\).
2. Given the polar equation \(r = 5 \csc \theta\), rewrite \(\csc \theta\) in terms of sine: \(\csc \theta = \frac{1}{\sin \theta}\), so the equation becomes \(r = \frac{5}{\sin \theta}\).
3. Multiply both sides of the equation by \(\sin \theta\) to get \(r \sin \theta = 5\).
4. Use the relationship \(y = r \sin \theta\) to substitute and rewrite the equation as \(y = 5\).
5. Interpret the rectangular equation \(y = 5\): this represents a horizontal line 5 units above the x-axis. In the polar coordinate system, this corresponds to all points where the vertical coordinate is 5.

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 interpreting graphs in both formats.

Conversion Formulas Between Polar and Rectangular Coordinates

Key formulas link polar and rectangular coordinates: x = r cos θ, y = r sin θ, and r² = x² + y². These allow transformation of equations from one system to the other, enabling easier analysis and graphing.

Trigonometric Functions and Their Reciprocal Identities

The equation involves csc θ, the reciprocal of sin θ (csc θ = 1/sin θ). Recognizing and manipulating these identities helps rewrite the polar equation in terms of x and y, facilitating conversion to rectangular form.