Problem 30 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 27-44: Find an equation in \(x\) and \(y\) that has the same graph as the polar equation. Use it to help sketch the graph in an \(r \theta\)-plane. $$ r=2 $$

Step-by-Step Solution

Answer

The equation in Cartesian coordinates is \( x^2 + y^2 = 4 \), which represents a circle with radius 2 centered at the origin.

Step 1: Understanding Polar Equation

The given polar equation is \( r = 2 \). In polar coordinates, \( r \) represents the distance from the origin to a point in the plane, and \( \theta \) is the angle formed with the positive x-axis. The equation \( r = 2 \) represents a circle centered at the origin with radius 2.

Step 2: Convert Polar Equation to Cartesian Equation

To convert the polar equation \( r = 2 \) to a Cartesian equation in terms of \( x \) and \( y \), use the relationships \( x = r \cos \theta \) and \( y = r \sin \theta \). Since \( r = \sqrt{x^2 + y^2} \), we can write the equation as \( \sqrt{x^2 + y^2} = 2 \). Squaring both sides gives \( x^2 + y^2 = 4 \).

Step 3: Analyze the Cartesian Equation

The equation \( x^2 + y^2 = 4 \) is the standard form of a circle in the Cartesian plane, centered at the origin with radius 2. This confirms the conversion is correct.

Step 4: Sketch the Graph

To sketch the graph of \( r = 2 \) in the \( r\theta \)-plane, draw a circle centered at the origin (0,0) with radius 2. Every point on the circle is at a constant distance of 2 from the origin regardless of the angle \( \theta \).

Key Concepts

Cartesian coordinates

Cartesian coordinates are a system of assigning ordered number pairs to points on a plane. This system uses two perpendicular axes that intersect at the origin, commonly known as the x-axis and the y-axis. Each point on the plane is defined by its horizontal position relative to the x-axis (known as the x-coordinate), and its vertical position relative to the y-axis (known as the y-coordinate).

• Cartesian coordinates provide a simple way to represent geometric figures such as lines, curves, and circles.
• The distance between two points in Cartesian coordinates can be calculated using the distance formula.

In this context, converting from polar to Cartesian coordinates helps us find a corresponding equation in the Cartesian plane, which is often easier to analyze and graph.

polar to Cartesian conversion

Converting polar coordinates to Cartesian coordinates involves changing the representation of a point from polar format, which uses the radius and angle, to Cartesian format, which uses x and y values. This conversion is based on trigonometric relationships:

• The x-coordinate is given by: \( x = r \cos \theta \).
• The y-coordinate is given by: \( y = r \sin \theta \).

To convert a polar equation like \( r = 2 \) into Cartesian coordinates, these formulas show that we can express the point's position in terms of x and y. Additionally, given that \( r = \sqrt{x^2 + y^2} \), we can simplify the expression further by squaring to eliminate the square root, resulting in an equation suitable for Cartesian plane analysis, such as \( x^2 + y^2 = 4 \). This technique allows us to view and sketch graphs more conveniently by utilizing familiar Cartesian plots.

circle equations

A circle can be described by a simple mathematical equation, especially in Cartesian coordinates. The standard form of a circle's equation is \( x^2 + y^2 = r^2 \), where \( (x, y) \) are the Cartesian coordinates of any point on the circle and \( r \) is the radius. For a circle centered at the origin, the equation simplifies, as there's no need for terms to account for shifts along the axes.

• This equation demonstrates that each point on the circle is equidistant from the center point (the origin).
• In our example, the equation \( x^2 + y^2 = 4 \) tells us the circle has a radius of 2 because \( r^2 = 4 \).

Understanding these equations makes it easier to visualize the shape and size of a circle, whether working in Cartesian coordinates or transitioning from a polar format.