Problem 81 - Algebra and Trigonometry Real Mathematics, Real People

Question

Describe the graph of the polar equation and find the corresponding rectangular equation. Sketch its graph. $$r=6$$

Step-by-Step Solution

Answer

The graph of the polar equation \(r = 6\) is a circle with a radius of 6 units centered at the origin. The corresponding rectangle equation is \(x^{2} + y^{2} = 36\). The graph is a circle that passes through the points (6,0), (-6,0), (0,6) and (0,-6).

Step 1: Describe the polar graph

The given polar equation is \(r = 6\). This represents a circle with radius 6 at the origin.

Step 2: Convert the polar equation to rectangular coordinates

The rectangle form of a polar equation is given by \(x^{2} + y^{2} = r^{2}\), where x and y are rectangular coordinates and r is the polar radius. In our case, r is 6, so the rectangular equation becomes \(x^{2} + y^{2} = 36\).

Step 3: Sketch the graph

A graph of the equation \(x^{2} + y^{2} = 36\) depicts a circle centered at the origin with a radius of 6 units. The circle will pass through the points (6,0), (-6,0), (0,6) and (0,-6) on the Cartesian plane.

Key Concepts

Polar Coordinates

Polar coordinates provide a method of representing points in a plane using a radius and an angle relative to a central point, often referred to as the pole (equivalent to the origin in rectangular coordinates). In this system, a point is defined as \( (r, \theta) \), where \r\ is the distance from the pole, and \theta\ is the angle measured from the positive x-axis (also called the polar axis) in a counterclockwise direction.

For example, \( r = 6 \) in polar coordinates describes a locus of points that lie 6 units from the pole, irrespective of the angle \theta\. This forms a circle with a constant radius, which is a fundamental concept when interpreting polar graphs as it translates to shapes and distances that we can easily visualize.

Rectangular Coordinates

Rectangular coordinates, also known as Cartesian coordinates, are arguably the most familiar coordinate system. They use two perpendicular axes, typically labeled x and y, intersecting at a point called the origin. Positions in this plane are given as ordered pairs \( (x, y) \), indicating horizontal and vertical displacements from the origin.

To connect the polar representation to the rectangular one, we use the conversion formulas \( x = r \cdot \cos(\theta) \) and \( y = r \cdot \sin(\theta) \). These allow us to translate a point given as a distance and angle into specific coordinates on the x and y axes, thus integrating the radial symmetry of polar equations with the grid structure of the rectangular coordinate system.

Circle Equations

The beauty of circle equations lies in their simplicity and symmetry. In polar form, a circle centered at the pole with radius \r\ is merely expressed as \( r = \text{constant} \). This equation tells us that no matter the angle \theta\, the radius is unchanging, resulting in a perfect circle.

In rectangular coordinates, the nuances of the equation become slightly more complex. Given a circle centered at the origin, the standard equation is \( x^{2} + y^{2} = r^{2} \), which equates the sum of the squares of x and y to the square of the radius. The conversion from polar to rectangular form for our given example, \(r = 6\), leads to \( x^{2} + y^{2} = 36 \) in rectangular form, epitomizing how the geometric elegance of circles is depicted algebraically in different coordinate systems.

Graphing Polar Equations

Graphing polar equations can be an enlightening display of patterns and symmetries that are less apparent in rectangular form. To graph a polar equation, one usually plots several points for various angles \theta\ and then connects them smoothly to reveal the curve. For instance, the polar equation \(r = 6\) is plotted by marking points on the plane that are 6 units away from the origin in any direction, forming a circle.

Unlike rectangular graphs, which are drawn on a grid, polar plots are typically sketched on a polar grid, where concentric circles represent distances from the pole, and radial lines indicate angles. Both types of graphs, however, provide valuable visual insight into the nature of the equations they represent, aiding in comprehension and application across mathematical disciplines.