Problem 59 - Algebra and Trigonometry

Question

Convert each polar equation to a rectangular equation. Then use a rectangular coordinate system to graph the rectangular equation. $$ r=8 $$

Step-by-Step Solution

Answer

The rectangular form of the polar equation \( r = 8 \) is \( x^{2} + y^{2} = 64 \). This represents a circle with a radius of 8 units in the rectangular coordinate system.

Step 1: Convert polar equation to rectangular form

The given equation is \( r = 8 \). In the rectangular coordinate system, the value of r is equal to the square root of the sum of x and y squared, that is: \( r = \sqrt{x^{2} + y^{2}} \). So we substitute \( \sqrt{x^{2} + y^{2}} \) for r in the equation, yielding: \( \sqrt{x^{2} + y^{2}} = 8 \). Squaring both sides of the equation, we get \( x^{2} + y^{2} = 64 \). So our rectangular equation is \( x^{2} + y^{2} = 64 \).

Step 2: Graphing the rectangular equation

Our rectangular equation, \( x^{2} + y^{2} = 64 \), represents a circle centered at the origin (0,0) with a radius of 8 units on the x-y plane. To graph this, we draw a circle with the center at the origin and a radius of 8 units. The circle will intersect the x-axis at the points (8,0) and (-8,0) and the y-axis at the points (0,8) and (0,-8).

Key Concepts

Rectangular Coordinates

Rectangular coordinates, also known as Cartesian coordinates, are a way to represent points on a plane using an ordered pair of numbers. Each point is defined by its distance from two perpendicular lines, the x-axis, and the y-axis.

To understand this system, think of a grid, like a piece of graph paper. The horizontal axis is the x-axis and the vertical axis is the y-axis. Each point on the plane is given by a pair of numbers

• The first number, the x-coordinate, tells you how far to move horizontally from the origin, which is the point (0,0).
• The second number, the y-coordinate, tells you how far to move vertically.

This system is fundamental for graphing equations, allowing us to visually represent mathematical relationships.

Coordinate Transformation

Coordinate transformation involves converting coordinates from one system to another. In our exercise, we changed a polar equation to a rectangular equation.

In polar coordinates, a point is determined by a distance from the origin (r) and an angle θ from the positive x-axis. To convert from polar to rectangular coordinates, we use these relationships:

• You can use the formula: \[ r^2 = x^2 + y^2 \]
• You can calculate x as: \[ x = r \cos(\theta) \]
• You can calculate y as: \[ y = r \sin(\theta) \]

For our specific case, where \( r = 8 \), we found that in rectangular coordinates, the equation is \( x^2 + y^2 = 64 \). This demonstrates that polar and rectangular coordinates are simply two ways of viewing the same points in a plane.

Equations of Circles

An equation of a circle in the rectangular coordinate system follows a specific form. The standard equation for a circle with a center at the origin (0,0) is:

• \[ x^2 + y^2 = r^2 \]

Where \( r \) is the radius of the circle. This form comes directly from the distance formula, which computes how far any point (x, y) is from the center of the circle.

In our problem, the circle equation \( x^2 + y^2 = 64 \) indicates:

• The circle is centered at the origin.
• It has a radius of 8, because \( \sqrt{64} = 8 \).

Understanding circle equations gives valuable insights into their size, location, and shape directly from their algebraic representation.

Graphing Equations

Graphing equations is the process of plotting points that satisfy the equation on a coordinate plane. This visual representation provides a clear picture of the relationship defined by the equation.

To graph a circle with the equation \( x^2 + y^2 = 64 \):

• Recognize it's a circle with its center at the origin (0,0).
• Determine the radius, \( r = 8 \), from the equation \( r^2 = 64 \).
• Plot the center of the circle, then use the radius to measure outwards in all directions to sketch the circle.
• This circle will intersect the axes at (8,0), (-8,0), (0,8), and (0,-8).

Graphing allows you to understand the spatial relation of mathematical concepts, confirming the link between algebraic equations and geometric figures.