Problem 50 - Precalculus Mathematics for Calculus

Question

Convert the polar equation to rectangular coordinates. $$r=-3$$

Step-by-Step Solution

Answer

The equation in rectangular coordinates is \(x^2 + y^2 = 9\).

Step 1: Understanding the Polar Equation

The given equation is a polar equation expressed in the form of \( r = -3 \). This indicates that every point on the graph is at a distance of -3 units from the origin. In polar coordinates, distance can't be negative, so we need to interpret this as moving 3 units in the opposite direction of the pole.

Step 2: Identifying the Polar Coordinates

The general form of a polar coordinate is \((r, \theta)\). For the equation \(r = -3\), we don't have a specific angle \(\theta\) given, so it could be any angle. This suggests a circle centered at the origin with radius 3, observed from the opposite side.

Step 3: Using the Relationship Between Polar and Rectangular Coordinates

Rectangular coordinates \((x, y)\) can be related to polar coordinates \((r, \theta)\) using the formulas: \( x = r \cos \theta \) and \( y = r \sin \theta \). These will help us convert the polar form to rectangular form.

Step 4: Expressing r in Rectangular Coordinates

To express \(r = -3\) in rectangular form, utilize the identity \( r = \sqrt{x^2 + y^2} \). So, set \(\sqrt{x^2 + y^2} = -3\), however, since \(r\) represents distance and is he ce positive, rewrite it as \(x^2 + y^2 = 9\). This represents a circle.

Step 5: Final Rectangular Form of the Equation

The equation \(x^2 + y^2 = 9\) is in rectangular coordinates and describes a circle centered at the origin with radius 3, which is the required conversion from the polar form \(r = -3\).

Key Concepts

Polar Coordinates

Polar coordinates are a way of representing points in a plane using a distance and an angle. In this system, each point is described by two values:

• \(r\): The radial coordinate, which represents the distance from the origin to the point.
• \(\theta\): The angular coordinate, which indicates the angle formed with the positive x-axis.

This system is particularly useful for problems involving circles and angles. The conversion of equations and points from polar to rectangular coordinates is a common task, as it allows different forms of representation based on the problem's requirements.

Rectangular Coordinates

Rectangular coordinates, or Cartesian coordinates, are the most familiar system for plotting points in a plane with the format

• \(x\): The horizontal distance from the origin, along the x-axis.
• \(y\): The vertical distance from the origin, along the y-axis.

This system is straightforward and ideal for operations involving algebra and geometry. Converting from polar to rectangular coordinates often requires using trigonometric relationships, as they provide a way to express the location of points and equations in a different mathematical form.

Polar Equations

Polar equations express relationships between the radial coordinate \(r\) and the angle \(\theta\). These equations can be effectively used to describe spiral shapes, circles, and other figures that are easily interpreted with angles and radii.
A typical example is the equation \(r = -3\), which needs interpretation because a negative radius in polar coordinates doesn’t have a straightforward meaning. It usually indicates a point that is directly opposite to what we might initially consider, requiring a conceptual adjustment to understand its implications in physical space.

Coordinate Transformation

Coordinate transformation is the process of converting between different types of coordinate systems, such as from polar to rectangular coordinates. For transformations:

• Use \(x = r \cos \theta\) to find the x-coordinate.
• Use \(y = r \sin \theta\) to find the y-coordinate.
• Utilize \(r = \sqrt{x^2 + y^2}\) to find the radial distance in polar from rectangular coordinates.

Understanding these transformations is key to solving many physics and calculus problems, as they allow for dynamic switching between systems to suit the needs of the analysis. For example, the transformation of \(r = -3\), results in the equation \(x^2 + y^2 = 9\) in rectangular coordinates, representing a circle despite the original indication of a negative radius.