Problem 48 - Thomas Calculus

Question

Replace the polar equations in Exercises \(27-52\) with equivalent Cartesian Replace the polar equations in Exercises \(27-52\) with equivalent Cartesian equations. Then describe or identify the graph. $$r=3 \cos \theta$$

Step-by-Step Solution

Answer

The Cartesian equation is \( x^2 + y^2 = 9 \), a circle centered at the origin with radius 3.

Step 1: Identify Given Polar Equation

The exercise asks us to convert the polar equation \(r = 3 \cos \theta\) into a Cartesian equation. In polar coordinates, \(r\) represents the radius from the origin to a point, and \(\theta\) is the angle with respect to the positive x-axis.

Step 2: Use Polar to Cartesian Conversion Formulas

Recall the conversion formulas between polar and Cartesian coordinates:- \(x = r \cos \theta\)- \(y = r \sin \theta\)- \(r^2 = x^2 + y^2\).Here, our goal is to eliminate \(r\) and \(\theta\) using these relations.

Step 3: Substitute for \(r \cos \theta\)

We begin by substituting \(x = r \cos \theta\) into the given equation. Since \(r = 3 \cos \theta\), multiply both sides of \(r \cos \theta = x\) by \(r\) to get:\[x = 3 r \cos \theta = 3x\]

Step 4: Solve for \(x\) in Terms of \(r\)

Given \(x = r \cos \theta\), substituting \(x = 3x\), which implies simplifying it results in\:\[r = 3 \left(\frac{x}{r}\right) = x\]Thus, simplifying gives you the equation of a vertical line where \(x = 3\).

Step 5: Convert \(r\) to Cartesian Form

By using the formula \(r^2 = x^2 + y^2\), we can square both sides of the equation \(x = r\) to get \(r^2 = 9\). So, \(x^2 + y^2 = 9\) which is the equation of a circle centered at the origin with radius 3.

Step 6: Identify the Graph

The Cartesian equation \( x^2 + y^2=9 \) corresponds to a circle centered at the origin with a radius of 3 units. This means every point satisfying this equation lies precisely 3 units from the origin, matching the original polar description.

Key Concepts

Polar Coordinates

Polar coordinates are a system used to locate points on a plane by specifying a distance and an angle. Think of this as using a radial path from a central point, known as the origin, to reach any given point. In polar coordinates, each point is defined by \( r \theta \ \), where \( r \) is the radius or the distance from the origin, and \( \theta \) is the angle formed with the positive x-axis (right on the plane). This method, much like using a compass, allows us to navigate easily along circular paths rather than straight lines.

• \( r \): distance from the origin
• \( \theta\): angle from the positive x-axis
• Direction and distance are fundamental in polar coordinates

Polar coordinates are especially useful in situations where dealing with circular or rotational motion is involved, like modeling the position of a point on a circular track. By simply adjusting \( \theta\) and \( r \), you can describe any point on the plane easily.

Cartesian Coordinates

Cartesian coordinates, unlike polar coordinates, are based on a grid or axes system that describes a point in a plane using \( x \) and \( y \) values. It’s much like reading a map and identifying a location with specific directions moving horizontally and vertically.

• \( x \): horizontal distance from the y-axis
• \( y \): vertical distance from the x-axis
• Uses a rectangular or grid layout

Conversion between polar and Cartesian coordinates involves specific formulas. You can transform polar coordinates to Cartesian by using:

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

This transformation helps in graphing equations more conveniently for straight lines and geometrical shapes, as seen in our exercise where the equation \( r = 3 \cos \theta \) is transformed into \( x^2 + y^2 = 9 \). This notably turns a simple polar equation into a Cartesian equation of a known shape—a circle.

Graph Identification

Identifying graphs from equations is all about recognizing patterns and forms that those equations represent visually. In the original exercise, we converted the polar equation \( r = 3 \cos \theta \) into the Cartesian equation \( x^2 + y^2 = 9 \). This new equation is easily identifiable as it matches the standard form of a circle centered at the origin.

• A circle has the form \( x^2 + y^2 = \text{radius}^2 \)
• This equation \( x^2 + y^2 = 9 \) means it's a circle with radius 3
• The center of the circle is at point \( (0,0) \)

By converting from one form of an equation to another, understanding the shape, size, and position on a graph becomes intuitive. Graph identification is crucial in mathematics and science as it helps visualize concepts for better analysis and understanding. By spotting these patterns quickly, one can infer properties and behaviors of mathematical models effectively.