In Exercises 59–74, convert each polar equation to a rectangular equation. Then use a rectangular coordinate system to graph the rectangular equation.r = 8

Verified step by step guidance

1. Start with the given polar equation: \( r = 8 \).
2. Recall the relationship between polar and rectangular coordinates: \( x = r \cos \theta \) and \( y = r \sin \theta \).
3. Since \( r = 8 \), substitute \( r \) in the equations for \( x \) and \( y \): \( x = 8 \cos \theta \) and \( y = 8 \sin \theta \).
4. Use the identity \( r^2 = x^2 + y^2 \) to convert the polar equation to a rectangular form: \( 8^2 = x^2 + y^2 \).
5. Simplify the equation to get the rectangular form: \( x^2 + y^2 = 64 \).

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 this system, a point is defined by the coordinates (r, θ), where 'r' is the radial distance and 'θ' is the angle. Understanding polar coordinates is essential for converting polar equations to rectangular form.

Rectangular Coordinates

Rectangular coordinates, also known as Cartesian coordinates, represent points in a plane using two perpendicular axes (x and y). The relationship between polar and rectangular coordinates is given by the equations x = r cos(θ) and y = r sin(θ). Converting polar equations to rectangular form involves using these relationships to express the equation in terms of x and y.

Graphing Equations

Graphing equations involves plotting points on a coordinate system to visualize the relationship defined by the equation. For polar equations converted to rectangular form, understanding how to interpret the resulting x and y values is crucial for accurately representing the graph. This process helps in identifying key features such as intercepts, symmetry, and overall shape of the graph.