Problem 53 - Thomas Calculus

Question

Replace the Cartesian equations in Exercises \(49-62\) by equivalent polar equations. $$ x^{2}+y^{2}=4 $$

Step-by-Step Solution

Answer

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

Step 1: Identify the Cartesian Coordinates

The Cartesian equation given is \(x^2 + y^2 = 4\). In the Cartesian coordinate system, this represents a circle with a radius of 2 centered at the origin (0,0).

Step 2: Conversion Formulas

Recall that the relationship between Cartesian and polar coordinates is given by the equations: \(x = r \cos \theta\), \(y = r \sin \theta\), and \(x^2 + y^2 = r^2\). Here, \(r\) is the radius, and \(\theta\) is the angle with the positive x-axis.

Step 3: Substitute Cartesian Terms with Polar Terms

Substitute \(x^2 + y^2\) with \(r^2\) in the given equation. Therefore, \(x^2 + y^2 = 4\) becomes \(r^2 = 4\).

Step 4: Solve for r

To find the polar form, solve for \(r\). Take the square root of both sides to obtain \(r = 2\).

Key Concepts

Cartesian Coordinates

Cartesian coordinates are a way to describe the position of a point in a two-dimensional plane using two values. These values are typically denoted by \(x\) and \(y\), which correspond to a point's horizontal and vertical distances from a fixed origin, usually located at the center of the plane. The point \( (x, y)\) signifies that the location is \(x\) units along the horizontal axis and \(y\) units along the vertical axis.
A few key points to remember about Cartesian coordinates are:

• The coordinate plane is divided into four quadrants, which help to easily identify the signs of \(x\) and \(y\).
• The origin, represented as \( (0, 0) \), is the center of the coordinate plane.
• Each point in the plane corresponds to one and only one pair of Cartesian coordinates.

Understanding Cartesian coordinates is crucial for graphing equations and understanding geometric shapes, like circles, which we often express using equations such as \(x^2 + y^2 = 4\). This equation describes a circle with a radius centered at the origin in Cartesian terms.

Circle Equation

The equation of a circle in the Cartesian coordinate system is typically written in the form \(x^2 + y^2 = r^2\), where \(r\) is the radius of the circle.
In the equation \(x^2 + y^2 = 4\), the number 4 represents the radius squared, indicating a circle with a radius of 2 units. Here’s why:

• The equation states that the sum of the squares of the distances from a point (x, y) on the circle to the origin is equal to 4.
• This equality \(x^2 + y^2 = 4\) implies that every point satisfying this condition is exactly 2 units away from the center (the origin), hence forming a circle.

Circles are symmetrical, and each point on the circle is equidistant from the center, which simplifies their representation by an equation. This form makes it easy to transition between different coordinate systems, like from Cartesian to polar.

Coordinate Transformation

Coordinate transformation is the process of converting one system of coordinates to another. This is particularly useful in mathematical computations where changing the perspective or simplifying equations can be advantageous.
When transitioning from Cartesian to polar coordinates:

• We use the formulas \(x = r \cos \theta\) and \(y = r \sin \theta\), which relate the horizontal and vertical positions to a radius \(r\) and an angle \( \theta\).
• The equation \(x^2 + y^2 = r^2\) helps equate Cartesian expressions directly to their polar counterparts.

These transformations often simplify the equations of geometric shapes. For instance, a circle in Cartesian coordinates, such as \(x^2 + y^2 = 4\), becomes a simpler polar equation \(r = 2\), easily interpreted as a circle centered at the origin with a constant radius of 2.
This makes polar coordinates particularly useful in scenarios involving rotations or circular motion, as they provide a more intuitive understanding of angles and distances.