91Ó°ÊÓ

Polar coordinates are a way to locate a point in a plane by specifying the distance from a reference point and an angle from a reference direction. The reference point, known as the pole, is analogous to the origin in Cartesian coordinates. The reference direction is usually the positive x-axis. Polar coordinates are represented as \( (r, \theta) \), where:

• \(r\) is the radial distance from the pole (origin).
• \(\theta\) is the angle measured from the positive x-axis in a counter-clockwise direction.

Polar coordinates are especially useful when dealing with circular or spiral patterns, as they simplify the representation of points on such curves. In the original exercise, the given equation \( r = 8 \sin \theta \) is expressed in polar form, indicating that the radius varies as a function of \( \theta \). To convert this to Cartesian coordinates, knowledge of both coordinate systems is essential.

Cartesian coordinates locate a point in a plane using two distances along orthogonal axes. These coordinates are represented as \((x, y)\), where:

• \(x\) is the horizontal distance from the vertical y-axis.
• \(y\) is the vertical distance from the horizontal x-axis.

The Cartesian coordinate system is convenient for many mathematical operations and is typically used when dealing with lines and shapes such as rectangles and squares. It requires transforming polar equations into Cartesian form to analyze in this context. In the exercise, converting the equation \( r = 8 \sin \theta \) into \( x^2 + y^2 = 8y \), subsequently leading to \( x^2 + (y-4)^2 = 16 \), utilizes the standard Cartesian representation to recognize familiar shapes.

Completing the square is a technique used to simplify quadratic equations and is particularly useful for rewriting equations in a form that reveals geometric properties more easily. The goal is to transform a quadratic into the form \((x-h)^2 + (y-k)^2 = r^2\), which is the standard form of a circle.

To complete the square for the term \(y^2 - 8y\) in the equation \(x^2 + y^2 - 8y = 0\), follow these steps:

• Take half of the linear coefficient of \(y\), which is \(-8\), resulting in \(-4\).
• Square this value to get \(16\).

Add and subtract \(16\) inside the equation to maintain equality, resulting in \(x^2 + (y-4)^2 = 16\). This presents a clear interpretation of the equation as a circle with center \((0, 4)\) and radius \(4\).

The equation of a circle in Cartesian coordinates is generally given by:

\[(x-h)^2 + (y-k)^2 = r^2\]
where:

• \((h, k)\) is the center of the circle.
• \(r\) is the radius of the circle.

In the context of the exercise, the equation \(x^2 + (y-4)^2 = 16\) represents a circle with a center at \((0, 4)\) and a radius of \(4\). This form is derived after completing the square, illustrating that what initially seems to be a complex relationship in polar form can be simplified to a familiar geometric shape in Cartesian form, allowing for easy graphing and understanding of the problem's geometric interpretation.