Problem 44 - Thomas Calculus

Question

Replace the polar equations in Exercises \(23-48\) by equivalent Cartesian equations. Then describe or identify the graph. $$ r=3 \cos \theta $$

Step-by-Step Solution

Answer

The equivalent Cartesian equation is \((x - \frac{3}{2})^2 + y^2 = \frac{9}{4}\); it's a circle centered at \((\frac{3}{2}, 0)\) with radius \(\frac{3}{2}\).

Step 1: Identify Polar Equation

The given polar equation is \( r = 3 \cos \theta \). This equation relates the radial coordinate \( r \) to the angle \( \theta \). Our task is to convert this into Cartesian coordinates \( (x, y) \).

Step 2: Use Polar to Cartesian Conversion Formulas

We use the formulas \( x = r \cos \theta \) and \( y = r \sin \theta \) to relate polar coordinates to Cartesian coordinates. We also have the relation \( r^2 = x^2 + y^2 \).

Step 3: Substitute and Simplify

Since \( r = 3 \cos \theta \), substitute \( \cos \theta = \frac{x}{r} \) into the equation: \( r = 3 \frac{x}{r} \). Rearranging gives \( r^2 = 3x \). Substituting \( r^2 = x^2 + y^2 \), we get \( x^2 + y^2 = 3x \).

Step 4: Rearrange into Standard Form

Rearrange the equation into standard conic section form: \( x^2 - 3x + y^2 = 0 \). Complete the square for the \( x \) terms: \( (x - \frac{3}{2})^2 \) to make it look like a circle equation. The equation becomes \( (x - \frac{3}{2})^2 + y^2 = \frac{9}{4} \).

Step 5: Identify the Graph

The equation \( (x - \frac{3}{2})^2 + y^2 = \frac{9}{4} \) is the equation of a circle. The circle has a center at \( (\frac{3}{2}, 0) \) and a radius of \( \frac{3}{2} \).

Key Concepts

Polar Coordinates

Polar coordinates are a way of representing points in a plane using a distance from a reference point and an angle from a reference direction. Instead of using traditional Cartesian coordinates, which specify a point using an x and y value, polar coordinates use:

• \( r \): The radial distance from the origin.
• \( \theta \): The angle measured from the positive x-axis in the counter-clockwise direction.

To visualize this, imagine you are at the center of a circle. The radial distance \( r \) tells you how far you are from the center, while the angle \( \theta \) directs you to the exact point around the circle. This coordinate system is particularly useful in scenarios where symmetry and circular paths are involved. Common examples include analyzing circular motion and waveforms.

Cartesian Coordinates

Cartesian coordinates use two perpendicular axes to locate a point in a plane. These axes, typically labeled as the x-axis and y-axis, form a grid that covers the entire plane with intersection points forming coordinates. Each point in this system is described by:

• \( x \): The horizontal position along the x-axis.
• \( y \): The vertical position along the y-axis.

For example, the point (3, 4) means starting at the origin (where both axes cross), you move 3 units along the x-axis and 4 units up along the y-axis. This system is versatile and intuitive for many types of problems, making it a foundation in algebra and geometry. It's often used when you need to define straight lines or when working with shapes such as rectangles and triangles.

Circle Equation

The equation of a circle is one of the simplest forms of conic sections and is expressed in Cartesian coordinates. Typically resolved around its center point, the standard formula for a circle is:\[(x - h)^2 + (y - k)^2 = r^2\]Where:

• \( (h, k) \): The center of the circle.
• \( r \): The radius of the circle.

In our solution, we converted the polar equation \( r = 3 \cos \theta \) into the Cartesian form of a circle \((x - \frac{3}{2})^2 + y^2 = \frac{9}{4}\). This shows that the circle is shifted 1.5 units to the right and resides on the origin along the y-axis. The circle's radius is \( \frac{3}{2} \), or 1.5 units. By completing the square, the equation highlights key features of the circle like its center and radius, making it simpler to visualize.

Conic Sections

Conic sections are curves obtained by intersecting a plane with a double-napped cone. These curves include various shapes such as ellipses, parabolas, hyperbolas, and circles. A circle is one such conic section when the plane cuts the cone parallel to its base.Each conic section has a standard Cartesian equation that helps identify its form:

• Circle: \((x-h)^2 + (y-k)^2 = r^2\)
• Ellipse: \( \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1 \)
• Parabola: \( y = ax^2 + bx + c \)
• Hyperbola: \( \frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1 \) or \( \frac{(y-h)^2}{a^2} - \frac{(x-k)^2}{b^2} = 1 \)

Understanding these forms helps in identifying and analyzing the graphs. Each shape has distinct geometric properties and real-world applications, such as orbits of planets (ellipses), satellite paths (parabolas), and optical systems (hyperbolas). Circles are most commonly found as wheels, ball-shaped objects and in many structural designs.