Problem 33 - Precalculus: Functions and Graphs

Question

Find a polar equation that has the same graph as the equation in \(x\) and \(y\). $$x^{2}+(y+3)^{2}=9$$

Step-by-Step Solution

Answer

The polar equation is \( r = 6 \sin(\theta) + 3 \).

Step 1: Convert Cartesian Equation to Standard Circle Form

The given equation is \( x^2 + (y+3)^2 = 9 \), which describes a circle in the Cartesian coordinate system. This equation can be rewritten as a standard circle equation: \((x - 0)^2 + (y - (-3))^2 = 3^2\). Thus, it represents a circle with center at \((0, -3)\) and radius \(3\).

Step 2: Express Circle Center in Polar Coordinates

The center of the circle in Cartesian coordinates, \((0, -3)\), can be converted to polar coordinates. In polar form, the center \((0, -3)\) has radial distance \(r_c = 3\) and angle \( \theta_c = \frac{3\pi}{2} \).

Step 3: Use Polar Circle Equation Formula

The polar equation of a circle with a center at \((r_c, \theta_c)\) and radius \(R\) is given by the formula: \( r = 2R \cos(\theta - \theta_c) + r_c \). Substituting the values \(R = 3\), \(r_c = 3\), and \(\theta_c = \frac{3\pi}{2}\), we have:\[r = 2 \times 3 \times \cos(\theta - \frac{3\pi}{2}) + 3.\]

Step 4: Simplify the Polar Equation

Compute \(\cos(\theta - \frac{3\pi}{2})\). Using the identity \(\cos(a - b) = \sin(b)\), we find \(\cos(\theta - \frac{3\pi}{2}) = \sin(\theta)\). Thus, the polar equation simplifies to: \[r = 6 \sin(\theta) + 3.\]

Key Concepts

Cartesian Coordinates

Cartesian coordinates are a system that uses two numerical values to determine a position on a plane. These two values are typically represented as

• x: the horizontal position on the plane
• y: the vertical position on the plane

Each point can be uniquely identified with a pair egin{equation}(x, y) ext{. } extFor example, the origin has coordinates } (0, 0). This system is beneficial for representing geometric shapes like lines, parabolas, and circles in a straightforward manner on a grid. When a circle equation is given as egin{equation} x^2 + (y - k)^2 = r^2 extnormal{, it signifies a circle centered at } (k) extnormal{ (with a radius } (r) extnormal {. ) Cartesian coordinates are instrumental because they provide a clear layout to manage various geometrical transformations within this plane, offering a simple way to visualize and solve problems related to distances and angles. }

Polar Coordinates

Polar coordinates offer an alternative means to describe locations on a plane.Instead of using horizontal and vertical distances, the polar coordinate system employs:

• r: the radial distance from the origin
• \(\theta\): the angle from the positive x-axis to the line connecting the origin to the point

For example, a point in polar coordinates is expressed as \((r, \theta)\). This approach is particularly handy for dealing with circular paths and periodic functions, as it emphasizes the radius and angular components.
In the context of a circle, this characteristic makes polar coordinates a suitable choice to depict curves efficiently, adapting perfectly to the circular symmetry. Simplifying trigonometric calculations further makes polar coordinates popular for scenarios involving rotational movement and cycles. For instance, any adjustment in the angle \(\theta\) allows us to easily rotate the point around the origin.

Circle Equations

Circle equations can be represented in either Cartesian or polar form, each offering a distinct perspective to understanding circles' geometry.
The standard Cartesian equation for a circle is \[(x - h)^2 + (y - k)^2 = r^2\]where \((h, k)\) is the circle's center and \(r\) its radius. This form effectively depicts the round shape using the distance formula.
In contrast, the polar equation for a circle offers a different visualization: \[r = 2R \cos(\theta - \theta_c) + r_c\] where

• \(R\) is the circle's radius,
• \(\theta_c\) is the angle corresponding to the center, and
• \(r_c\) is the radial distance to the center from the origin

These polar circle equations simplify scenarios where understanding the angle and radial distance is more straightforward, emphasizing the circular symmetry inherent in the shape.

Coordinate Transformation

Coordinate transformation is the process that allows conversion between different coordinate systems, such as from Cartesian to polar coordinates and vice versa. This transformation is essential because each system has specific uses or advantages depending on the problem context. For instance:

• Cartesian coordinates are often preferred for linear movement analysis
• Polar coordinates often excel in rotational or radial symmetry contexts

The transformations between the systems include:

• Converting from Cartesian to Polar: \(r = \sqrt{x^2 + y^2}\) and \(\theta = \tan^{-1}(y/x)\)
• Converting from Polar to Cartesian: \(x = r \cos(\theta)\) and \(y = r \sin(\theta)\)

By enabling these switches, coordinate transformations reveal different insights or simplifications, making problem-solving more flexible and efficient in areas ranging from physics to engineering and computer graphics.