Problem 22 - Precalculus Mathematics for Calculus

Question

Sketch a graph of the polar equation, and express the equation in rectangular coordinates. $$r=\cos \theta$$

Step-by-Step Solution

Answer

Convert to \((x - \frac{1}{2})^2 + y^2 = \frac{1}{4}\), a circle with radius \(\frac{1}{2}\).

Step 1: Understanding Polar Coordinates

In polar coordinates, the point is represented by \((r, \theta)\), where \(r\) is the radial distance from the origin and \(\theta\) is the angle from the positive x-axis. The given equation \(r = \cos \theta\) is polar because it describes \(r\) as a function of the angle \(\theta\).

Step 2: Converting to Rectangular Coordinates

To convert polar equations to rectangular form, use the relationships: \(x = r \cos \theta\), \(y = r \sin \theta\), and \(r^2 = x^2 + y^2\). The given \(r = \cos \theta\) can be rewritten by replacing \(\cos \theta = \frac{x}{r}\) since \(x = r \cos \theta\). Therefore, the equation becomes \(r = \frac{x}{r}\), implying \(r^2 = x\). Since \(r^2 = x^2 + y^2\), substituting gives \(x^2 + y^2 = x\).

Step 3: Rewriting the Rectangular Equation

Starting from \(x^2 + y^2 = x\), rearrange it as a conic section equation: \(x^2 - x + y^2 = 0\). Completing the square on \(x\): \((x - \frac{1}{2})^2 - \frac{1}{4} + y^2 = 0\). This simplifies to \((x - \frac{1}{2})^2 + y^2 = \frac{1}{4}\), which represents a circle with center \(\left(\frac{1}{2}, 0\right)\) and radius \(\frac{1}{2}\).

Step 4: Sketching the Graph

In polar coordinates, \(r = \cos \theta\) describes a circle passing through points where the angle \(\theta\) results in positive \(r\). With our rectangular form \((x - \frac{1}{2})^2 + y^2 = \frac{1}{4}\), we can draw this circle centered at \(\left(\frac{1}{2}, 0\right)\) with radius \(\frac{1}{2}\). Mark this circle on a coordinate plane to complete the sketch.

Key Concepts

Converting Polar to Rectangular Coordinates

Polar coordinates provide a different system of locating points on a plane by using a radius and an angle from a reference direction like the positive x-axis. To convert these into the more familiar rectangular (or Cartesian) coordinates, we utilize a few key relationships derived from trigonometry. These relationships are:

• For the x-coordinate: \[ x = r \cos \theta \]
• For the y-coordinate: \[ y = r \sin \theta \]
• For the relationship involving both coordinates: \[ r^2 = x^2 + y^2 \]

Given a polar equation like \( r = \cos \theta \), substituting \( \cos \theta \) with \( \frac{x}{r} \) based on the relationship \( x = r \cos \theta \) allows us to translate it into the equation \( r = \frac{x}{r} \). Solving for \( r^2 \), we derive \( r^2 = x \). This is a pivotal step in transforming the equation into rectangular coordinates.

Equation of a Circle

An equation of a circle in rectangular coordinates is generally expressed as \[ (x - h)^2 + (y - k)^2 = r^2 \] where \((h, k)\) is the center of the circle, and \(r\) is its radius.
When dealing with the equation obtained from polar coordinates, we often end up with expressions like \(x^2 + y^2 = x\).
This equation is reshaped to fit a standard form by completing the square, which will reveal the circle's center and radius. In our example, rewriting it as \((x - \frac{1}{2})^2 + y^2 = \frac{1}{4}\) indicates a circle centered at \( ( \frac{1}{2}, 0) \) with a radius given by the square root of \( \frac{1}{4} \), which is \( \frac{1}{2} \). This understanding helps in visualizing and sketching the circle accurately on a graph.

Completing the Square

Completing the square is a useful algebraic method for converting a quadratic equation into a form that makes it easier to analyze or use, particularly in defining shapes like circles on a graph. Here's how it works:

• Start with the equation: \(x^2 - x + y^2 = 0\)
• Focus on the terms involving \(x\): \(x^2 - x\).
• Reformat these terms into a perfect square trinomial: Add and subtract \( \left(\frac{1}{2}\right)^2 \). Now it looks like: \((x - \frac{1}{2})^2 - \frac{1}{4} \).
• Balance the equation: Move \(- \frac{1}{4}\) to the right side, giving you \((x - \frac{1}{2})^2 + y^2 = \frac{1}{4}\).

This transformation clarifies the equation's geometric interpretation, showing it represents a circle. By completing the square, we identify our circle's center and radius, which greatly assists in graphing or further analyzing the geometric object. This technique is especially potent when dealing with conic sections, such as parabolas, ellipses, and hyperbolas, not just circles.