Question

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

Short answer

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

Step-by-step solution

Introduction to Polar Coordinates

Polar coordinates express a point in terms of a distance from the origin and an angle \( \theta \) from the positive x-axis. The conversion formulas are: \( x = r \cos \theta \) and \( y = r \sin \theta \).

Substitute Rectangular Coordinates

Given the equation \( 3x - y = 2 \), substitute \( x = r \cos \theta \) and \( y = r \sin \theta \) to transform it into polar coordinates.

Transform Equation

Substitute to get: \( 3(r \cos \theta) - (r \sin \theta) = 2 \). Simplify to obtain: \( r(3 \cos \theta - \sin \theta) = 2 \).

Solve for \( r \)

Isolate \( r \) by dividing both sides by \( 3 \cos \theta - \sin \theta \): \[ r = \frac{2}{3 \cos \theta - \sin \theta} \].

Consider Graphing the Equation

The polar equation \( r = \frac{2}{3 \cos \theta - \sin \theta} \) describes a line in the polar coordinate system. To sketch it, consider that any specific \( \theta \) leads to a specific \( r \), generating points along this line.

Key concepts

Conversion Formulas

Polar and rectangular coordinates are two different ways of describing the same location on a plane. Conversion formulas help translate between these two systems. The formulas are:

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

To convert a rectangular equation (which uses \(x\) and \(y\)) to a polar equation (which uses \(r\) and \(\theta\)), substitute \( x = r \cos \theta \) and \( y = r \sin \theta \) into the equation. These substitutions change all parts of the equation into terms involving \(r\) and \(\theta\), which define a point's distance and angle from the origin, respectively. This transformation is essential for analyzing equations within the polar coordinate system.

Rectangular Coordinates

Rectangular coordinates are the most common way to represent points on a two-dimensional plane, using an \(x\)-coordinate and a \(y\)-coordinate.

• The \(x\)-coordinate measures the distance horizontally from the origin.
• The \(y\)-coordinate measures the distance vertically from the origin.

For example, in the original equation \(3x - y = 2\), you see these coordinates in action. By interpreting the equation as a line in the rectangular coordinate system, it's easy to plot using the slope-intercept form: \(y = mx + b\). However, to analyze and graph using polar coordinates, conversion is necessary, turning the equation into a polar form that can display its characteristics differently on a polar plane.

Polar Equations

Polar equations represent relationships where points are described in terms of distance from the origin and angle \(\theta\). This system rotates around a point 'O' or the origin.
An intermediary step involves transforming a rectangular equation, like \(3x - y = 2\), into a polar form. This begins by substituting with conversion formulas: \(3(r \cos \theta) - (r \sin \theta) = 2\).
After simplifying, you achieve a clear representation \(r(3 \cos \theta - \sin \theta) = 2\), which provides insight into how a line is configured in polar form. Finally, solving for \(r\) gives:

• \( r = \frac{2}{3 \cos \theta - \sin \theta} \)

Using an equation in this way reveals how the distance \(r\) changes with the angle \(\theta\), defining a path that seems different when viewed in polar coordinates.

Graphing Polar Equations

Graphing polar equations begins with interpreting the expressions for \(r\) in relation to \(\theta\). With \(r = \frac{2}{3 \cos \theta - \sin \theta}\), the graph translates numerical outputs for \(r\) based on varied \(\theta\) inputs.
To draw this:

• Select several \(\theta\) values. Often angles like \(0\), \(\frac{\pi}{4}\), and \(\frac{\pi}{2}\) provide insightful points.
• Calculate corresponding \(r\) for these angles using the polar equation.
• Plot each point \((r, \theta)\).

Connecting these points reveals the graph's shape. Unlike rectangular graphs emphasizing lines and curves, polar graphs highlight rotational symmetry and distance. Understanding these patterns guides better insight into functions and mapping within circular environments.