Problem 44 - Precalculus: Functions and Graphs

Question

Find an equation in \(x\) and \(y\) that has the same graph as the polar equation. Use it to help sketch the graph in an \(r \theta\) -plane. $$r-6 \cos \theta=0$$

Step-by-Step Solution

Answer

The Cartesian equation is \((x-3)^2 + y^2 = 9\), a circle centered at (3,0) with radius 3.

Step 1: Understanding the Polar Equation

The given polar equation is \( r - 6 \cos \theta = 0 \). To find an equivalent equation in Cartesian coordinates \(x\) and \(y\), we need to express \(r\) and \(\theta\) in terms of \(x\) and \(y\).

Step 2: Convert Polar to Cartesian Coordinates

Recall the relationships: \( r = \sqrt{x^2 + y^2} \), \( \cos \theta = \frac{x}{r} \). Substitute these into the polar equation: \( \sqrt{x^2 + y^2} = 6\frac{x}{\sqrt{x^2 + y^2}} \).

Step 3: Solve for Cartesian Equation

Multiply both sides by \(\sqrt{x^2 + y^2}\) to eliminate the radical: \(x^2 + y^2 = 6x\).

Step 4: Complete the Square

To complete the square, rearrange the equation: \(x^2 - 6x + y^2 = 0\). Complete the square for \(x\): \((x-3)^2 - 9 + y^2 = 0\).

Step 5: Final Cartesian Equation

The equation becomes \((x-3)^2 + y^2 = 9\), representing a circle with center (3,0) and radius 3.

Key Concepts

Polar Equations

Polar equations are expressed in terms of the variables \( r \) and \( \theta \), where \( r \) is the distance from the origin, and \( \theta \) is the angle from the positive x-axis. They are particularly useful for describing curves and shapes where the relation to the origin is a key component, such as spirals or circles. Unlike Cartesian equations, polar forms can often describe certain shapes more simply. For example, the equation \( r = 6 \cos \theta \) defines a circle in the polar coordinate system. Converting polar equations to Cartesian form can clarify relationships and allow for easier graphing, especially if you are more familiar with the Cartesian system.

To understand the graph of a polar equation like \( r = 6 \cos \theta \), it is helpful to know how the polar plane works. The "spokes" represent the angles, while circles (or arcs) represent the different values of \( r \). This grid helps visualize the effects of changing \( \theta \) on \( r \). Therefore, knowing how to manipulate polar equations and convert them can open up a deeper understanding of the geometry involved.

Cartesian Coordinates

The Cartesian coordinate system uses the \( x \) and \( y \) axes to represent points and describe equations in a two-dimensional space. In this system, each point is defined by a pair of coordinates \( (x, y) \), mapping positions on the grid. This method is straightforward for plotting lines, parabolas, circles, and many other geometric shapes.

Converting from polar to Cartesian coordinates involves using two primary relationships:

• \( r = \sqrt{x^2 + y^2} \)
• \( \cos \theta = \frac{x}{r} \)

These conversions allow us to take polar equations, which relate angle and distance from the origin, and rewrite them in terms of \( x \) and \( y \). Therefore, once converted, these equations often represent well-known shapes more familiar to those used with the Cartesian system, like lines or circles. The conversion process also aids in sketching graphs when switching between the systems.

Completing the Square

Completing the square is a technique used to simplify quadratic expressions or equations, particularly helpful in deriving standard forms. It is especially useful in transforming the general form of a circle's equation to its standard form.

For an equation like \( x^2 - 6x + y^2 = 0 \), completing the square involves:

• Rearranging the terms to group \( x \)-terms together.
• Adding and subtracting a number to form a perfect square trinomial.

In this scenario, we focus on the \( x \)-terms: \( x^2 - 6x \). We find the square by taking half the coefficient of \( x \), \( -3 \), then squaring it, resulting in \( 9 \), which then transforms the expression to \( (x-3)^2 \). Thus, the adjusted equation becomes \( (x-3)^2 + y^2 = 9 \), which effectively simplifies analyzing and graphing this equation.

Circle Equation

A circle in the Cartesian plane is defined by its center and radius. The most basic form of a circle's equation is \( (x-h)^2 + (y-k)^2 = r^2 \), where \((h, k)\) is the center of the circle, and \(r\) is its radius.

When converting a polar equation, like \( r = 6 \cos \theta \), to its Cartesian equivalent, completing the square helps identify this circle form. The final equation \( (x-3)^2 + y^2 = 9 \) indicates a circle centered at \((3,0)\) with a radius of 3, aligning with the general circle equation form. This clear identification allows for straightforward graphing and geometric insight, thus simplifying visual representations of these equations in both their polar and Cartesian forms. Managing these conversions efficiently empowers understanding and interpretation of geometric properties.