Problem 57 - Thomas Calculus

Question

Replace the Cartesian equations in Exercises \(53-66\) with equivalent polar equations. $$x^{2}+y^{2}=4$$

Step-by-Step Solution

Answer

The polar equation is \( r = 2 \).

Step 1: Review Polar Coordinate Definitions

Remember that in polar coordinates, the point \(x, y\) can be represented as \(r, heta\), where \(x = r \cos(\theta)\) and \(y = r \sin(\theta)\). Additionally, \(r^2 = x^2 + y^2\).

Step 2: Use the Polar Coordinate Formula

Given the equation \(x^{2}+y^{2}=4\), use the polar coordinate relation \(r^2 = x^2 + y^2\). Substitute this to convert the Cartesian equation to polar form, yielding \(r^2 = 4\).

Step 3: Simplify the Polar Equation

The equation \(r^2 = 4\) can be simplified to \(r = 2\) or \(r = -2\), though the radius \(r\) is typically considered non-negative, so the principal solution is \(r = 2\).

Key Concepts

Cartesian to Polar Conversion

Converting between Cartesian and polar coordinates is a common task in mathematics, particularly when dealing with circular shapes and motions. In the Cartesian system, points are defined using an ordered pair \((x, y)\), whereas the polar system uses a radius \(r\) and an angle \(\theta\) from the positive x-axis.
The basic formulas for conversion are straightforward:

• \(x = r \cos(\theta)\)
• \(y = r \sin(\theta)\)

To find \(r\), which is the distance from the origin to the point, use: \[r = \sqrt{x^2 + y^2}\]
For \(\theta\), which is the angle formed with the positive x-axis, you can calculate:

• \(\theta = \arctan\left(\frac{y}{x}\right)\) for typical scenarios
• Use adjustments if \(x\) or \(y\) values are zero, or if the point lies on an axis.

These conversions allow for easier computations and transformations, especially in problems involving rotations or circular paths.

Polar Equations

Polar equations, unlike their Cartesian counterparts, describe the relationship between the radius \(r\) and the angle \(\theta\). These equations are particularly useful for problems involving circles, spirals, and oscillations.
An example of a basic polar equation is \(r = 2\), describing a circle of radius 2 centered at the origin. This equation implies that no matter the angle, the point will always be 2 units away from the origin.
Working with polar equations often involves identifying how changes in \(\theta\) affect \(r\) to form geometric shapes. Calculations in polar coordinates can simplify complex geometry and trigonometry problems by focusing only on radii and angles. Polar graphs also unveil symmetries and rotations easily, aiding visual understanding of the problem's inherent properties.

Coordinate Systems

Coordinate systems are frameworks used to determine the location of points in a space. Each system is designed for specific types of spaces or problems, such as linear, circular, or multi-dimensional spaces.

• Cartesian Coordinates: Used for linear positioning and rectangular grids. The system consists of perpendicular axes (x and y in 2D) intersecting at the origin.


• Polar Coordinates: Ideal for circular and rotational contexts. Points are located using a distance (radius) and an angle, making them great for dealing with periodic functions and circular motions.


• Other Systems: Such as cylindrical and spherical, exist to handle 3D problems and more complex shapes.

Each coordinate system has advantages in specific scenarios, and selecting the right one can simplify calculations and enhance understanding of the problem at hand. Mastering the transition between these systems allows for flexibility and efficiency in problem-solving.