Problem 47 - Algebra and Trigonometry Real Mathematics, Real People

Question

Convert the rectangular equation to polar form. Assume \(a<0\) $$x^{2}+y^{2}=9$$C

Step-by-Step Solution

Answer

The polar form of the given rectangular equation \(x^{2}+y^{2}=9\) is \(r^{2}=9\).

Step 1: Identify given Rectangular coordinates

The given rectangular(coordinate) equation is \(x^{2}+y^{2}=9\).

Step 2: Apply fundamental Polar-Rectangular relationship

In polar representation, \(r = \sqrt{x^{2}+y^{2}}\), and the square of that is \(r^{2} = x^{2}+y^{2}\). We can substitute \(x^{2}+y^{2} = r^{2}\) in our given equation as it's equivalent to that form.

Step 3: Substitution

Substituting \(x^{2}+y^{2}\) with \(r^2\), the equation becomes \(r^{2} = 9\).

Key Concepts

Rectangular Coordinates

Rectangular coordinates, often referred to as Cartesian coordinates, are a fundamental way of pinpointing position in a 2D space. These coordinates use two axes, typically labeled as 'x' and 'y', which intersect at a point called the origin. Every location in this coordinate system is determined by a pair of numerical values known as an ordered pair. For example, the point \((3, 4)\) indicates a position 3 units along the x-axis, and 4 units along the y-axis. Rectangular coordinates are particularly useful for describing geometrical shapes, equations, and graphs.• **Axes:** Defined by horizontal (x-axis) and vertical (y-axis) lines.• **Origin:** The point where the axes cross, denoted as \((0,0)\).Expressions like \(x^2 + y^2 = 9\) use rectangular coordinates to describe geometric figures. In this example, the equation outlines a circle with a radius of 3 units, centered at the origin. Understanding this system forms the basis for converting between coordinate types, such as shifting from rectangular to polar coordinates.

Equation Conversion

Equation conversion, specifically between rectangular and polar forms, is a significant mathematical process. It helps in simplifying complex problems by switching between two different perspectives of coordinate representation. In the example \(x^2 + y^2 = 9\), conversion to a polar form involves recognizing the equivalent expressions for rectangular coordinates using polar elements.• **Polar Coordinates Association:** The polar equivalent of the expression involves the radius \(r\), and the connections are derived from basic trigonometry.• **Key Relationship:** The relationship \(r^2 = x^2 + y^2\) is pivotal for the conversion process.In polar coordinates, we express points in terms of the angle \(\theta\) and the radius \(r\). By substituting \(x^2 + y^2\) with \(r^2\), we translate our equation into polar terms, resulting in \(r^2 = 9\). This highlights the circle's radius in the polar coordinate system, maintaining the geometric integrity of shapes across transformations.

Mathematical Transformation

Mathematical transformation refers to the method of changing an equation from one form into another while preserving its essential properties. The transformation from rectangular to polar coordinates involves using known relationships between these systems. It shifts focus from the x and y axes to a radial distance and angle configuration.• **Fundamental Formula:** Use \(r = \sqrt{x^2 + y^2}\) to move into the polar system.• **Implication of Transformation:** The polar form of an equation can make it simpler to understand certain attributes of a graph, such as symmetry and periodicity.In the current example, transforming \(x^2 + y^2 = 9\) into polar form achieves a simpler expression \(r^2 = 9\), which exactly represents the circle's constant radius. This transformation integrates geometry and algebra to provide a new perspective, facilitating the solution of problems involving circles and other shapes.