Problem 33 - Thomas Calculus

Question

Replace the polar equations in Exercises \(23-48\) by equivalent Cartesian equations. Then describe or identify the graph. $$ r=\frac{5}{\sin \theta-2 \cos \theta} $$

Step-by-Step Solution

Answer

The polar equation represents the straight line \( y = 2x + 5 \).

Step 1: Understand Polar to Cartesian Conversion

In polar coordinates, each point is determined by the radius \( r \) and angle \( \theta \). We can convert these points into Cartesian coordinates \((x, y)\) using the formulas \( x = r \cos \theta \) and \( y = r \sin \theta \). Also, note that \( \sin \theta = \frac{y}{r} \) and \( \cos \theta = \frac{x}{r} \).

Step 2: Substitute Polar Formulas

Rewrite the polar equation \( r = \frac{5}{\sin \theta - 2 \cos \theta} \) using \( \sin \theta = \frac{y}{r} \) and \( \cos \theta = \frac{x}{r} \). Substitute these into the equation: \[ r = \frac{5}{\frac{y}{r} - 2 \cdot \frac{x}{r}} \]

Step 3: Simplify the Equation

Multiply both sides by the denominator \((\frac{y}{r} - 2 \cdot \frac{x}{r})\) to eliminate it:\[ r \left(\frac{y}{r} - 2 \cdot \frac{x}{r}\right) = 5 \]The \( r \)'s cancel out, resulting in: \[ y - 2x = 5 \].

Step 4: Convert to Cartesian Form

The final result from simplifying is a linear equation: \( y - 2x = 5 \), which is already in Cartesian coordinates. This represents the form \( y = mx + b \) of a straight line, where \( m = 2 \) and \( b = 5 \).

Step 5: Identify the Graph

The Cartesian equation \( y = 2x + 5 \) represents a straight line with a slope of 2 and a y-intercept of 5. This line will have a consistent increase in \( y \) as \( x \) increases, with no curves or deviations.

Key Concepts

Polar Equations

Polar equations are a way to represent mathematical relationships in terms of a radius and an angle. They are used widely in mathematics and physics because they can simplify the representation of circular and rotational paths.
In a polar coordinate system, each point is described by \( r \) (the distance from the origin) and \( \theta \) (the angle from the positive x-axis).

• The polar coordinate system is handy for equations involving angles, rotations, or circles.
• Common conversions are often needed to translate these into Cartesian coordinates to graph or analyze them further.
• Polar equations can sometimes appear complex but are solvable and interpretable in Cartesian form.

Understanding these transformations from polar to Cartesian is beneficial for solving problems that involve geometric shapes or directional components.

Cartesian Coordinates

Cartesian coordinates are a fundamental way to represent positions on a plane using horizontal and vertical axes, typically labeled as \( x \) and \( y \). They make it easier to describe points and shapes on a flat surface.

• In a 2D coordinate plane, every point is defined by a pair of numbers \((x, y)\).
• These coordinates help simplify the process of plotting graphs and solving geometric problems.
• Converting from polar to Cartesian coordinates involves using the relationships \( x = r \cos \theta \) and \( y = r \sin \theta \).

The Cartesian system is straightforward for analyzing and visualizing linear relationships, as exemplified in the equation derived from the given polar equation.

Linear Equations

Linear equations are mathematical expressions that represent a straight line when graphed on a coordinate plane, characterized by the formula \( y = mx + b \). They play a crucial role in algebra and calculus.

• In this form, \( m \) is the slope of the line - it shows how steep the line is.
• \( b \) is the y-intercept - it indicates where the line crosses the y-axis.
• A linear equation ensures a constant rate of change, making the graph a straight line without curves.

For example, from the polar conversion steps, the equation \( y - 2x = 5 \) was derived and rearranged into \( y = 2x + 5 \).
This describes a straight line with a slope of 2 and a y-intercept of 5, clearly identifying a rescaled straight path in a Cartesian coordinate system.