Problem 31 - Thomas Calculus

Question

Replace the polar equations in Exercises \(23-48\) by equivalent Cartesian equations. Then describe or identify the graph. $$ r^{2}=1 $$

Step-by-Step Solution

Answer

The Cartesian equation is \( x^2 + y^2 = 1 \), which is a circle with center at the origin and radius 1.

Step 1: Understand the Polar Equation

The given polar equation is \( r^2 = 1 \). In polar coordinates, \( r \) represents the distance from the origin to a point, and \( \theta \) is the angle from the positive x-axis. Our task is to express this equation in Cartesian coordinates.

Step 2: Convert Polar to Cartesian Coordinates

The formulas for conversion from polar to Cartesian coordinates are: \( x = r \cos(\theta) \) and \( y = r \sin(\theta) \). Since \( r^2 = x^2 + y^2 \), we can use this to find the equivalent Cartesian equation.

Step 3: Substitute and Simplify

From Step 2, we use the relationship \( r^2 = x^2 + y^2 \). Substituting the given equation \( r^2 = 1 \), we get \( x^2 + y^2 = 1 \). This is the Cartesian form of the given polar equation.

Step 4: Identify the Graph

The equation \( x^2 + y^2 = 1 \) represents a circle in the Cartesian plane. The circle is centered at the origin (0,0) with a radius of 1.

Key Concepts

Polar Coordinates

Polar coordinates are a method of plotting points on a plane using a combination of distance and angle. Instead of using the traditional x and y coordinates, polar coordinates use

• \( r \) which represents the radial distance from the origin
• \( \theta \) which is the angle measured from the positive x-axis

This system is particularly useful when dealing with scenarios involving circular or spiral patterns. For example, in polar coordinates, the point \((r, \theta)\) can directly describe a point's location in terms of its distance from the center and its direction. This allows for more intuitive manipulation of objects in a rotational setting. Converting these into Cartesian terms often helps when performing calculations since Cartesian expressions such as algebraic equations are more widely used.

Cartesian Coordinates

Cartesian coordinates are the most familiar system of coordinates, characterized by using x and y values to specify a point's position on a two-dimensional plane. Each point in this system is identified by

• \( x \) which indicates the horizontal distance from the origin
• \( y \) which represents the vertical distance from the origin

This coordinate system is rectangular, meaning coordinates run along perpendicular axes. This makes it ideal for computing linear equations and analyzing many types of geometric shapes. Because of its widespread use, many complex equations in other coordinate systems are often transformed into Cartesian coordinates to simplify calculations and graphical representations.

Circle Equation

A circle in Cartesian coordinates is conveniently expressed with the equation format \( x^2 + y^2 = r^2 \), where

• \( x \) and \( y \) are the coordinates of any point on the circle
• \( r \) is the radius of the circle

This equation highlights a set of all points that maintain a consistent distance from a central point. Thus, for a standard circle, \((x - h)^2 + (y - k)^2 = r^2\), where \((h, k)\) are the coordinates of the circle's center. The simplicity of this representation allows easy analysis of various properties of a circle, such as its symmetry and its boundary limits. When a polar equation \(r^2 = 1\) is converted, it becomes \(x^2 + y^2 = 1\), describing a circle centered at the origin with a radius of one unit.

Geometric Transformations

Geometric transformations in mathematics involve changing the position, size, or shape of a figure. Common transformations include

• Translation, which slides a figure from one position to another
• Rotation, which turns a figure around a fixed point
• Reflection, which flips a figure over a line
• Scaling, which changes the size of a figure while keeping its shape

When converting coordinates, such as from polar to Cartesian, you effectively perform a type of transformation, changing the framework in which a figure is described without altering the shape itself. It is key to understand how these adjustments affect not only the object but its respective equation. For example, a circle remains a circle when seen through the lens of a different coordinate system, demonstrating the flexibility and interconnectivity of mathematical representations.