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. Start with the given polar equation: \(r = 5 \csc \theta\).
2. Recall that \(\csc \theta = \frac{1}{\sin \theta}\), so rewrite the equation as \(r = \frac{5}{\sin \theta}\).
3. Multiply both sides by \(\sin \theta\) to eliminate the fraction: \(r \sin \theta = 5\).
4. Use the identity \(r \sin \theta = y\) to convert to rectangular coordinates, giving \(y = 5\).
5. Recognize that \(y = 5\) is a horizontal line in the rectangular coordinate system, which corresponds to a circle centered at the origin with radius 5 in the polar coordinate system.

Key Concepts

Polar Coordinates

Polar coordinates represent points in a plane using a distance from a reference point (the origin) and an angle from a reference direction (usually the positive x-axis). In polar coordinates, a point is expressed as (r, θ), where 'r' is the radial distance and 'θ' is the angle. Understanding this system is crucial for converting polar equations to rectangular form.

Conversion from Polar to Rectangular Coordinates

To convert polar equations to rectangular form, we use the relationships x = r cos(θ) and y = r sin(θ). These equations relate the polar coordinates to the Cartesian coordinates (x, y). For the given equation r = 5 csc(θ), recognizing that csc(θ) = 1/sin(θ) allows us to manipulate the equation into a rectangular format.

Graphing Polar Equations

Graphing polar equations involves plotting points based on their polar coordinates and understanding how these points relate to the Cartesian plane. After converting to rectangular form, one can identify key features such as intercepts and asymptotes, which aid in sketching the graph accurately. Familiarity with the shapes of common polar graphs, like circles and lines, enhances this process.