Problem 61 - Precalculus Mathematics for Calculus

Question

Convert the polar equation to rectangular coordinates. $$r=\frac{1}{\sin \theta-\cos \theta}$$

Step-by-Step Solution

Answer

The equation in rectangular coordinates is \( \sqrt{x^2 + y^2}(y - x) = 1 \).

Step 1: Identify the Polar Equation

The given polar equation is \( r = \frac{1}{\sin \theta - \cos \theta} \). Our goal is to convert this equation from polar form (\( r, \theta \)) to rectangular form (\( x, y \)).

Step 2: Express Polar Coordinates in Terms of Rectangular Coordinates

In rectangular coordinates, \( x = r \cos \theta \) and \( y = r \sin \theta \). Also, \( \sin \theta = \frac{y}{r} \) and \( \cos \theta = \frac{x}{r} \).

Step 3: Substitute Trigonometric Expressions

Substitute \( \sin \theta \) and \( \cos \theta \) with their respective expressions in terms of \( x \), \( y \), and \( r \). The equation becomes \( r = \frac{1}{\frac{y}{r} - \frac{x}{r}} = \frac{r}{y-x} \).

Step 4: Solve for \( r(y-x) \)

Multiply every term by \( y-x \) to clear the denominator: \( r = \frac{r}{y-x} \Rightarrow r(y-x) = 1 \).

Step 5: Substitute \( r \) with \( \sqrt{x^2 + y^2} \)

Since \( r = \sqrt{x^2 + y^2} \) in rectangular coordinates, substitute \( r \) in the equation: \( \sqrt{x^2 + y^2}(y-x) = 1 \).

Step 6: Final Expression in Rectangular Coordinates

The rectangular form of the given polar equation is \( \sqrt{x^2 + y^2}(y - x) = 1 \).

Key Concepts

Rectangular Coordinates

Rectangular coordinates are a way to locate a point in a plane using two numerical values, typically denoted as \(x\) and \(y\). They form part of the Cartesian coordinate system, where \(x\) represents the horizontal position and \(y\) represents the vertical position of a point. This coordinate system is extensively used for graphing equations on a two-dimensional plane.

To convert a point from polar to rectangular coordinates, we use the relations:

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

Here, \(r\) is the radial distance from the origin, and \(\theta\) is the angle made with the positive x-axis. By applying these equations, any polar coordinate point, defined by \(r\) and \(\theta\), can be translated into rectangular coordinates comprising \(x\) and \(y\).

Understanding rectangular coordinates is crucial as they provide a direct method for calculating distances and angles in physics and engineering contexts.

Trigonometric Substitution

Trigonometric substitution is a mathematical technique used to simplify complex expressions by replacing trigonometric functions with equivalent forms. In converting polar equations to rectangular coordinates, this method helps in transforming the terms involving \( \sin \theta \) and \( \cos \theta \) into expressions involving \(x\), \(y\), and \(r\).

For example, we know:

• \( \sin \theta = \frac{y}{r} \)
• \( \cos \theta = \frac{x}{r} \)

In the exercise, substituting these expressions into the original polar equation \( r = \frac{1}{\sin \theta - \cos \theta} \), results in the equation \( r = \frac{r}{y-x} \.\) This form allows further simplification, eventually transforming the equation into a completely rectangular format.

Trigonometric substitution not only aids in such conversions but also plays a pivotal role in solving integrals and differential equations, where recognition of such equivalences can significantly streamline calculations.

Coordinate Transformation

Coordinate transformation is the process of converting one form of coordinates into another, often used to simplify equations or to analyze them in a more convenient form.

In the given problem, the goal is to transform a polar equation into a rectangular form. This requires applying the relationships:

• \( x = r \cos \theta \)
• \( y = r \sin \theta \)
• \( r = \sqrt{x^2 + y^2} \)

The transformation is executed by re-expressing the polar coordinates in terms of \(x\) and \(y\). In this case, after simplifying the equation with trigonometric substitutions, the transformation results in \( \sqrt{x^2 + y^2}(y - x) = 1 \).

Coordinate transformation not only facilitates simplification of equations but also enhances visualization and understanding of geometric problems. It's a key tool in many areas of mathematics and physics, aiding in transition from one coordinate system to another based on convenience and requirement.