Problem 37 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 27-44: 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. $$ r(\sin \theta-2 \cos \theta)=6 $$

Step-by-Step Solution

Answer

The equivalent Cartesian equation is \( y = 2x + 6 \).

Step 1: Convert Polar to Rectangular Coordinates

To change from polar to rectangular coordinates, use the conversions: \(x = r\cos\theta \) and \( y = r\sin\theta \). The \( r \( \sin\theta - 2 \cos\theta \) \) from the given equation becomes \( y - 2x = 6 \) when expanded.

Step 2: Isolate Terms

Express the equation in standard form for a straight line by isolating the terms. From \((y - 2x) = 6 \), keep y on one side: \( y = 2x + 6 \), indicating this is a linear equation.

Step 3: Sketch the Graph

To sketch the graph in the \( r\theta \)-plane, since \( r\theta \) corresponds to a line in the Cartesian \( x\, y \)-plane, plot the line \( y = 2x + 6 \) using points like \( (0, 6) \) and \( (3, 12) \). Convert these points back to polar to assist in plotting in the polar plane.

Key Concepts

Polar Coordinates

Polar coordinates describe a point in a plane using a distance from a reference point and an angle from a reference direction. This system uses the notation \( (r, \theta) \) where:

• \( r \) represents the radial distance from the origin to the point.
• \( \theta \) measures the angle from the positive x-axis to the line segment connecting the origin to the point.

This system is analogous to the way a compass uses direction and distance. Polar coordinates are especially useful in situations involving angles and circles, like when dealing with periodic functions or circular motion. Conversion between polar and rectangular coordinates is possible using the formulas:

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

This conversion allows for easier graphing in the Cartesian plane and provides an alternative way of representing geometric equations.

Rectangular Coordinates

Rectangular coordinates (also known as Cartesian coordinates) represent a point's position in a plane with an ordered pair \( (x, y) \). This coordinate system is laid out on a grid defined by the x-axis and y-axis. Here,

• \( x \) indicates horizontal distance from the origin.
• \( y \) shows vertical distance.

Understanding and plotting these coordinates are crucial for graphing equations and visualizing functional relationships. Rectangular coordinates offer a straightforward way to describe straight lines by using linear equations, as they provide clear visualization of slopes and intercepts. Graphically, the rectangular coordinate system helps in determining intersection points, symmetries, and other features of algebraic curves. This system is particularly effective in analyzing complex equations once they have been transformed from polar form.

Linear Equations

Linear equations describe straight lines and have the general form \( y = mx + b \). Here,

• \( m \) represents the slope, indicating the steepness or inclination of the line.
• \( b \) is the y-intercept, the point where the line crosses the y-axis.

Linear equations are fundamental because they form the basis of more complex algebraic structures and are used widely across different fields including physics, economics, and engineering. By converting polar equations to rectangular form, these lines can often be more easily understood and graphed. In our context, the polar equation \( r(\sin\theta - 2\cos\theta)=6 \) was converted to the linear equation \( y = 2x + 6 \). This conversion facilitates sketching the graph by pinpointing intersections like \( (0, 6) \) and understanding the line's behavior. Having both polar and rectangular views of the equation provides a comprehensive toolset for analyzing mathematical problems.