Question

Write each rectangular equation in polar form. $$x^{2}+y^{2}=1$$

Short answer

\(r^{2} = 1\)

Step-by-step solution

Review the Relationship Between Rectangular and Polar Coordinates

Understand the conversion formulas between rectangular and polar coordinates: For a point in the plane, rectangular coordinates \(x, y\) can be transformed to polar coordinates \(r, \theta\) using the equations \(x = r \cos\theta\) and \(y = r \sin\theta\). The equation \(r^{2} = x^{2} + y^{2}\) relates the distance from the origin to the point in both coordinate systems.

Identify the Given Rectangular Equation

Recognize the given rectangular equation is \(x^{2}+y^{2}=1\).

Substitute the Polar Formulas

Substitute the expressions for \(x\) and \(y\) in rectangular coordinates with their corresponding expressions in polar coordinates. Replace \(x\) with \(r\cos(\theta)\) and \(y\) with \(r\sin(\theta)\) in the equation.

Simplify the Polar Equation

Square both \(r\cos(\theta)\) and \(r\sin(\theta)\), and sum them to replace \(x^{2}+y^{2}\), yielding \(r^{2}(\cos^{2}(\theta) + \sin^{2}(\theta))\). Using the Pythagorean identity \(\cos^{2}(\theta) + \sin^{2}(\theta) = 1\), simplify the equation to \(r^{2} = 1\).

Key concepts

Polar Coordinates

Polar coordinates are an alternative to rectangular (also known as Cartesian) coordinates for representing points in a two-dimensional plane. Unlike the rectangular system, which uses horizontal and vertical components, polar coordinates describe a point based on its distance from a reference point (called the pole, or origin) and the angle it forms with a reference direction (usually the positive x-axis). A point in polar coordinates is expressed as a pair \(r, \theta\), where \(r\) is the radial distance from the origin, and \(\theta\) is the angular component, typically measured in radians.

In context to our equation \(x^{2}+y^{2}=1\), the polar form will capture the essence of this circle with its center at the origin and a radius of 1. It showcases the elegant simplicity of polar coordinates in representing circular shapes and patterns.

Rectangular Coordinates

Rectangular coordinates, also referred to as Cartesian coordinates, are the most common coordinate system used to represent points in a two-dimensional space. In this system, a point's location is determined by an ordered pair \(x, y\), where \(x\) is the horizontal distance from the vertical y-axis, and \(y\) is the vertical distance from the horizontal x-axis. These perpendicular axes divide the plane into four quadrants.

The equation \(x^{2}+y^{2}=1\) in rectangular coordinates defines a circle with a radius of 1 centered at the origin of the coordinate plane. This is a prime example of how quadratic expressions in rectangular coordinates can represent various conic sections and geometric shapes, offering a clear visual interpretation of algebraic equations.

Pythagorean Identity

The Pythagorean identity is a cornerstone of trigonometry and is derived from the Pythagorean theorem. It states that for any angle \(\theta\), the sum of the squares of the sine and cosine of that angle is always equal to one: \(\cos^{2}(\theta) + \sin^{2}(\theta) = 1\). This fundamental relationship between sine and cosine provides an invaluable tool for simplifying trigonometric expressions and converting between trigonometric forms.

When solving the exercise, applying the Pythagorean identity simplifies the polar equation and is crucial in transforming the rectangular coordinate equation \(x^{2}+y^{2}=1\) into its polar form. This step demonstrates the seamless interplay between algebraic identities and coordinate transformations.

Coordinate Transformation

Coordinate transformation is the process of converting the representation of points from one coordinate system to another. The transformation between rectangular and polar coordinates is especially important in fields like physics and engineering, where different problems are more naturally addressed within a particular coordinate system.

In transforming the given rectangular equation \(x^{2}+y^{2}=1\) into polar form, we use the relationship between the two systems, with \(x = r \cos(\theta)\) and \(y = r \sin(\theta)\). This conversion leverages the inherent structure of both systems to express the same geometric information in a more contextually relevant form. Understanding these transformations enhances a student's flexibility and problem-solving skills by providing multiple perspectives on how to approach and interpret equations in different settings.