Problem 110 - Precalculus with Limits

Question

In Exercises 109-118, describe the graph of the polar equation and find the corresponding rectangular equation. Sketch its graph. \(r=8\)

Step-by-Step Solution

Answer

The graph of the polar equation \(r=8\) is a circle centered at the origin with radius 8. The corresponding rectangular equation is \((x^2 + y^2) = 64\).

Step 1: Identify the polar graph

The graph of a polar equation of the form \(r=a\) where \(a\) is a constant, is a circle centered at the origin of radius \(a\). Hence, the graph of the polar equation \(r=8\) is a circle of radius 8 centered at the origin.

Step 2: Translate to Rectangular Coordinates

To convert this polar equation to rectangular form, remember that in polar coordinates, the radial distance \(r\) equals the square root of the sum of the squares of the x and y coordinates of the equivalent rectangular point. Therefore, we have the equation \((x^2 + y^2) = (r^2)\), which leads to \((x^2 + y^2) = 64\).

Step 3: Sketch the Graph

Now that we know the equation in both polar and rectangular forms, we can sketch the graph. In a polar coordinate system, this will be a circle with radius 8 centered at the origin. In a rectangular coordinate system, this will again be a circle, but now centered at the origin (0, 0), with radial lines marking distance 8 from the origin in all directions.

Key Concepts

Polar Equations

In mathematics, polar equations provide a unique method for describing positions and shapes based on the distance from a central point (radius, r) and an angle from a reference direction (theta, \( \theta \)). A simple polar equation like \( r = a \) represents a circle with radius \( a \) centered at the origin in a polar coordinate system.

For instance, when we encounter an equation such as \( r = 8 \), it describes a circle where any point on the circle is 8 units away from the central point, or origin. The beauty of polar equations lies in their ability to describe complex curves and shapes, especially those with a circular or spiral nature, in a more straightforward and intuitive way compared to rectangular equations.

The polar coordinate system can be especially useful in fields such as physics and engineering where rotations and circular motions are commonly analyzed. However, envisioning polar equations can be challenging; graph sketching becomes an invaluable tool to visualize the shapes they represent.

Rectangular Equations

In contrast to polar equations, rectangular equations use the familiar Cartesian coordinate system (x and y axes) to describe geometric shapes. The conversion from polar equations, such as \( r = 8 \), to rectangular equations often involves trigonometric identities and the Pythagorean theorem.

To transform the given simple polar equation into its rectangular counterpart, we utilize the relationships \( x = r\cdot\cos\theta \) and \( y = r\cdot\sin\theta \), yielding \( x^2 + y^2 = 64 \) for the circle of radius 8. This equation captures all the points that are exactly 8 units away from the origin in any direction, hence forming a circle in the rectangular coordinate system.

Understanding these equations allows students to switch between coordinate systems, depending on which is more convenient for the problem at hand. Mastery of this concept is crucial for higher-level mathematics, as it lays the foundation for analyzing the geometry and algebra of curves.

Graph Sketching

Graph sketching is a fundamental skill that aids in understanding and interpreting mathematical concepts. It is the visual representation of equations on a coordinate plane. In the context of the problem \( r = 8 \), graph sketching enables students to visualize the circle defined by the polar equation.

In a polar coordinate system, sketching involves plotting points at various angles from the origin and measuring the radial distance from the origin out to the points. The circular shape comes together when points at all angles from 0 to 360 degrees are at an equal distance from the center.

Improving Sketch Accuracy

When sketching the graph of \( x^2 + y^2 = 64 \) in rectangular coordinates, precision is achieved by plotting key points (such as the intersections with the axes, at (8,0), (0,8), (-8,0), and (0,-8)) and ensuring the curve is symmetrical about both the x and y axes. Software or graphing calculators can enhance this process, but the fundamental skill of manual graph sketching reinforces a student's understanding of the equation's implications.