Problem 161 - Calculus Volume 3

Question

Convert the rectangular equation to polar form and sketch its graph. $$ 3 x-y=2 $$

Step-by-Step Solution

Answer

The polar form is \( r = \frac{2}{3\cos\theta - \sin\theta} \).

Step 1: Understand the Rectangular Equation

The given rectangular equation is in the form of a linear equation:\[ 3x - y = 2 \]To convert this to polar form, we need to express both \( x \) and \( y \) in terms of \( r \) and \( \theta \). Remember, \( x = r\cos\theta \) and \( y = r\sin\theta \).

Step 2: Substitute Polar Coordinates

Substitute \( x = r \cos \theta \) and \( y = r \sin \theta \) into the rectangular equation:\[ 3(r \cos \theta) - (r \sin \theta) = 2 \]

Step 3: Factor Out the Common Factor

Factor out \( r \) from the equation:\[ r (3\cos\theta - \sin\theta) = 2 \]

Step 4: Solve for r

To express the equation entirely in terms of polar coordinates, solve for \( r \):\[ r = \frac{2}{3\cos\theta - \sin\theta} \]

Step 5: Sketch the Graph

To sketch the graph, consider various values for \( \theta \) (e.g., \( 0, \frac{\pi}{4}, \frac{\pi}{2}, \pi \)) and find corresponding values for \( r \). Plot these points in polar coordinates and join them to form a line. This will give a line that extends radially based on the angle \( \theta \).

Key Concepts

Rectangular to Polar Conversion

Converting from rectangular coordinates, which use the coordinate pair \((x, y)\), to polar coordinates, which rely on the pair \((r, \theta)\), is a key skill in mathematics. Rectangular coordinates describe a point's location in a plane using horizontal and vertical distances, whereas polar coordinates use distance from the origin and angle from the positive x-axis.
To perform this conversion, remember the relationships:

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

Thus, each coordinate is expressed in terms of \(r\) and \(\theta\).
For a given equation like \(3x - y = 2\), you substitute \(x\) and \(y\) with these polar equivalents. The equation becomes \(3(r \cos \theta) - (r \sin \theta) = 2\).
By factoring out \(r\), we simplify it further to \(r = \frac{2}{3\cos \theta - \sin \theta}\).
This expression in polar form is now ready to work with in various applications, including graphing.

Linear Equation Graphing

Graphing linear equations helps to visually understand relationships between variables. In most cases, we are looking to find the line's location and slope. Linear equations like \(3x - y = 2\) can also be examined in polar form, as seen with the conversion to \(r = \frac{2}{3\cos\theta - \sin\theta}\).
Start by considering specific angles (or \(\theta\) values), such as \(0\), \(\frac{\pi}{4}\), and \(\pi\). For each angle, calculate the corresponding \(r\) to determine the length from the origin.
Once you have various points, you can plot them on a polar graph. The points join to illustrate the line.
While it might look different compared to a rectangular plot, pattern regularities like straight lines remain identifiable, though they might appear as curves based on perspective in polar graphs.

Polar Equations

Polar equations are a way to represent lines and curves using the polar coordinate system. They provide a flexible approach to understanding equations in a radial framework instead of the typical horizontal and vertical setup.
The polar form \(r = \frac{2}{3\cos\theta - \sin\theta}\) shows how we can represent a linear relationship in polar coordinates. It involves expressing the radial distance \(r\) as a function of the angle \(\theta\).
With polar equations, angle \(\theta\) plays a crucial role in shaping the representation of the graph. Measurement in radians often helps easy interpretation, with values recalling cardinal directions relative to the polar origin.
This approach can be especially useful for calculations involving circular or spiral patterns, like waveforms or navigation paths, where directions and distances from a central point are more critical than specific up/down or left/right displacements.