Problem 114 - Precalculus with Limits

Question

In Exercises 109-118, describe the graph of the polar equation and find the corresponding rectangular equation. Sketch its graph. \(r=4\ \cos\ \theta\)

Step-by-Step Solution

Answer

The graph of the polar equation \(r = 4\cos\theta\) is a circle centered at (2,0) with radius 2. The rectangular form of the equation is \(x^2 - 4x + y^2 = 0\).

Step 1: Understanding the Polar Equation

The polar equation given in this problem is \(r=4\cos\theta\). This equation represents a circle with radius 2 and center at (2,0) in polar coordinates.

Step 2: Convert to Rectangular Coordinates

To convert from polar to rectangular coordinates, use the equations \(x = r\cos\theta\) and \(y = r\sin\theta\). Starting from the polar equation \(r = 4\cos\theta\), and multiplying both sides of the equation with \(r\) gives \(r^2 = r \cdot 4\cos\theta\), which is equivalent to \(x^2 + y^2 = 4x\). Rearranging it finally gives the rectangular form of the equation as \(x^2 - 4x + y^2 = 0\).

Step 3: Sketch the Graph

Sketch a circle centered at (2,0) with radius 2. Make sure to label the center and a few points on the circle to show its size and location. Plot the positive x-axis from the center to the right most part of the circle since in polar coordinates, \(r\) is always positive.

Key Concepts

Polar Coordinates

In mathematics, polar coordinates provide a system where each point on a plane is defined by an angle and a distance from a reference point. The reference point is known as the pole, and the radial coordinate is the distance between the point and the pole.
Think of this system like a radar—you locate points not by moving horizontally or vertically (as in rectangular coordinates), but rather by indicating how far out you look and at what angle.
The angles are typically measured in radians or degrees, originating from a fixed direction known as the polar axis, which is usually the positive x-axis. This system is best used for graphs of circles, spirals, and other shapes that are symmetric about the origin.

Rectangular Coordinates

Rectangular coordinates, also known as Cartesian coordinates, describe points using a pair of numerical values. They are plotted along the horizontal x-axis and vertical y-axis.
Key advantages of rectangular coordinates include how they naturally map to square grids, making calculation and visualization straightforward for many purposes.

• The origin in rectangular coordinates, where both x and y values are zero, is the starting point.
• Each point is defined by a horizontal and vertical shift from this origin.

This system suits problems involving grid-based locations like city streets or computer screens.

Converting Polar Equations

Converting polar equations to rectangular form involves using relationships between the systems:

• The equation for converting the radial coordinate is given by \(x = r\cos\theta\) and \(y = r\sin\theta\).
• Conversely, you can express \(r\) through \(r = \sqrt{x^2 + y^2}\), and \(\tan\theta = \frac{y}{x}\).

For example, to convert the polar equation \(r = 4\cos\theta\) to rectangular coordinates, multiply both sides by \(r\): 1. Start with \(r^2 = r\cdot4\cos\theta\).2. Use \(r^2 = x^2 + y^2\) to transform it into \(x^2 + y^2 = 4x\).3. Rearrange for the Cartesian form: \(x^2 - 4x + y^2 = 0\).
These steps bridge the gap between the two systems, enabling easy graph sketching.

Graphing Polar Equations

Graphing polar equations involves plotting specific points that satisfy the equation and then connecting them to visualize the entire graph.
In the equation \(r = 4\cos\theta\), the graph represents a circle. Here's how to sketch it:

• Identify that each solution \(r\) and \(\theta\) maps to a unique point on the plane.
• Determine the center of the circle is at (2,0) with a radius of 2.
• Plot major points on the circle:
• (0, 0) when \(\theta = 90°\) or \(270°\) (\(r = 0\)).
• (2, 2) when \(\theta = 0°\) (maximum radius \(r = 4\)).

Ensuring accuracy here facilitates a clear, understandable visual representation mapping the polar function to familiar Euclidean shapes.