Problem 42 - Precalculus: Functions and Graphs

Question

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 graph is a circle centered at the origin with radius 2.

Step 1: Convert Polar to Cartesian

The polar equation is given as \(r = 2\). We need to convert this to a Cartesian equation. In polar coordinates, \(r\) represents the radius, and \(\theta\) represents the angle. The relationships between the polar and Cartesian coordinates are given by \(x = r \cos \theta\) and \(y = r \sin \theta\). Thus, the equation becomes \(x^2 + y^2 = r^2\). Substituting \(r = 2\) gives \(x^2 + y^2 = 4\).

Step 2: Identify the Graph Type

The equation \(x^2 + y^2 = 4\) represents a circle with a center at the origin \((0, 0)\) and a radius of 2. This indicates that the graph is a circle centered at the origin with a radius of 2 in the Cartesian \(xy\)-plane.

Step 3: Sketch the Graph

To visualize the equation in the \(r\theta\)-plane, note that the equation \(r = 2\) describes a circle where the distance from the origin \(r\) is constant and equal to 2 for all angles \(\theta\). Hence, in the polar plane, this is a circle centered at the origin with radius 2.

Key Concepts

Polar Coordinates

Polar coordinates are a way of representing points in a plane using a distance and an angle. The point is located by determining two values:

• \( r \), the radial distance from the origin (often written as \( r \)), and
• \( \theta \), the angle from the positive x-axis.

The polar coordinate system is very useful for problems involving symmetry around a point, such as circles or spirals. In a polar coordinate system, each point is identified uniquely by the pair \((r, \theta)\). In the example given, the equation \( r = 2 \) indicates that each point is 2 units away from the origin, forming a perfect circle around it.
This system is particularly handy when dealing with circular and rotational patterns, allowing for simpler equations and graphs.

Cartesian Coordinates

Cartesian coordinates, also known as rectangular coordinates, are expressed as an ordered pair \((x, y)\) in a two-dimensional plane. Each value represents a position from a predefined zero point, called the origin, where the x-axis and y-axis intersect.

• The horizontal axis is known as the x-axis, and
• the vertical axis is known as the y-axis.

In the Cartesian system, the equation \( r^2 = x^2 + y^2 \) is used to relate it to polar coordinates.
This transformation allows us to convert polar equations like \( r = 2 \) into Cartesian forms\( x^2 + y^2 = 4 \), making it easy to visualize geometric shapes such as lines and curves on the familiar grid.
An understanding of both systems is crucial for more complex math problems that require transitioning between different types of graphs and forms.

Circle Graphs

Circle graphs are a common occurrence in both polar and Cartesian coordinate systems. Circles, being symmetrical and consistent in distance from a center point, are easily expressed in polar coordinates as \( r = 2 \). This indicates that the radius is consistent, resulting in a circle with constant radius.
In Cartesian coordinates, a circle centered at the origin is represented by the equation \( x^2 + y^2 = r^2 \). For the given conversion \( x^2 + y^2 = 4 \), it signifies a circle with a radius of 2. These insights help us understand how circles maintain their general shape and properties regardless of the system used.
Moreover, understanding these representations aids in graphing circles properly across different planes, ensuring consistency in math and real-life applications.