Problem 31 - A Graphical Approach to Precalculus with Limits

Question

For each rectangular equation, give its equivalent polar equation and sketch its graph. $$x^{2}+y^{2}=16$$

Step-by-Step Solution

Answer

Equivalent polar equation is \( r = 4 \), representing a circle with radius 4 centered at the origin.

Step 1: Recall Polar Coordinates Definitions

The polar coordinate system represents points using the distance from the origin and the angle from the positive x-axis. The key relationships are: \(x = r \cos \theta\) and \(y = r \sin \theta\), where \(r\) is the radius and \(\theta\) is the angle.

Step 2: Substitute Polar Equations

Replace \(x\) and \(y\) in the equation \(x^2 + y^2 = 16\) using the polar substitutions. This gives us: \( (r \cos \theta)^2 + (r \sin \theta)^2 = 16 \).

Step 3: Simplify the Equation

Expand and simplify the expression: \( r^2 \cos^2 \theta + r^2 \sin^2 \theta = 16 \). Using the identity \( \cos^2 \theta + \sin^2 \theta = 1 \), we simplify further to get \( r^2 = 16 \).

Step 4: Solve for the Polar Equation

Take the square root of both sides of the equation: \( r = \sqrt{16} \), resulting in \( r = 4 \). Thus, the equivalent polar equation is \( r = 4 \).

Step 5: Sketch the Polar Graph

In the polar coordinate system, \( r = 4 \) represents a circle centered at the origin with a radius of 4 units. Draw a circle around the origin with 4 units as the radius.

Key Concepts

Rectangular Equations

Rectangular equations are the most common way to express mathematical relationships in the Cartesian coordinate system. This system describes points by their horizontal (x) and vertical (y) displacements from the origin. In the given problem, we have the rectangular equation \(x^2 + y^2 = 16\). This equation defines a circle:

• Centered at the origin (0, 0)
• With a radius of 4

To visualize this, imagine a point that is always 4 units away from the center. As you fix this distance and rotate around the origin, you form a perfect circle.
It is useful to convert these equations into other forms such as polar equations to simplify analysis or graphing of the shapes especially when symmetry is involved, as in circles.

Polar Equations

The conversion from rectangular to polar coordinates provides a more intuitive understanding for certain types of graphs, such as circles or spirals.
In polar coordinates, a point is described by two values: the radius \(r\) (distance from the origin) and the angle \(\theta\) (direction from the positive x-axis).
This system is ideal when dealing with circular motions or rotations around the origin. For our equation, \(x^2 + y^2 = 16\), we translate this to a polar equation using:

• \(x = r \cos \theta\)
• \(y = r \sin \theta\)

After substitution and simplification, we find the polar form \(r = 4\).
This indicates all points at a radius of 4 from the origin, confirming our equation forms a circle centered at the origin with radius 4.

Graph Sketching

Sketching graphs in polar coordinates may initially seem challenging but can be quite direct with practice.
For the polar equation \(r = 4\), the graph is a circle where every point has the same distance, 4 units, from the center.
Here’s how to sketch it:

• Start at the origin, which is the center of your polar plot.
• Measure out 4 units (or the equivalent scale) in all directions, outlining the radius.
• Draw smooth arcs connecting these points, forming a perfect circle.

In a Cartesian grid, our initial equation \(x^2 + y^2 = 16\) represented this same circle.
In both systems, the symmetry and distance constraints help maintain the shape, making transformations between coordinate systems practical and visually intuitive.