Problem 40 - Contemporary Precalculus

Question

Find a polar equation that is equivalent to the given rectangular equation. $$y=x-2$$

Step-by-Step Solution

Answer

Question: Given the rectangular equation y=x-2, find the equivalent polar equation. Answer: The equivalent polar equation is $$r=\frac{-2}{sin(\theta)-cos(\theta)}$$.

Step 1: Write down the given rectangular equation

We are given the rectangular equation: $$y=x-2$$

Step 2: Substitute the expressions for x and y in polar coordinates

Replace x with $$rcos(\theta)$$ and y with $$rsin(\theta)$$ in the given equation: $$rsin(\theta)=rcos(\theta)-2$$

Step 3: Simplify the equation

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

Step 4: Isolate r

Divide both sides of the equation by $$(sin(\theta)-cos(\theta))$$ to isolate r: $$r=\frac{-2}{sin(\theta)-cos(\theta)}$$ So, the equivalent polar equation is: $$r=\frac{-2}{sin(\theta)-cos(\theta)}$$.

Key Concepts

Rectangular to Polar Conversion

Converting rectangular equations to polar form involves translating a system based on Cartesian coordinates into one using polar coordinates. In rectangular coordinates, a point in the plane is described by \(x, y\). In polar coordinates, a point is described using \(r\) (the distance from the origin) and \(\theta\) (the angle from the positive x-axis). This conversion is essential for solving problems involving circular and rotational symmetries, which are inherently simpler in polar terms.

### Common Substitutions
To perform a rectangular to polar conversion, you should know these key substitutions:

• \(x = r\cos(\theta)\)
• \(y = r\sin(\theta)\)
• \(r^2 = x^2 + y^2\) (although not used in this problem, it's useful for others)

For our problem, the original rectangular equation is \ y = x - 2 \.

Substitute \(x\) with \(r\cos(\theta)\) and \(y\) with \(r\sin(\theta)\), resulting in:

• \(r\sin(\theta) = r\cos(\theta) - 2\)

This equation forms the basis for further simplification of the polar form.

Trigonometric Substitution

Trigonometric substitution plays a vital role in converting and simplifying trigonometric equations. Once we've substituted the polar relations into the rectangular equation, using trigonometric identities and algebra helps manipulate the equation into a more recognizable form.

### Understanding the EquationGiven the equation after substitution: \(r\sin(\theta) = r\cos(\theta) - 2\), it's important to manage the terms efficiently. This involves grouping them to isolate specific components.

### FactorizationIn step 3 of the solution, factor out \(r\) such that:

• \(r(sin(\theta) - cos(\theta)) = -2\)

Factorization simplifies the manipulation of the equation and sets up the final step of simplification, showing how subtraction of trigonometric functions can be handled similarly to linear terms. Using trigonometric identities effectively is important even if the identity itself isn't simplified further in this problem.

Equation Simplification

Simplification of equations, especially in polar form, is essential for making the relationship between variables clear and concise. In polar coordinates, variables are often intertwined through trigonometric functions.

### Isolating rIn the final step, isolate \(r\) by dividing both sides of the equation by the combined trigonometric function \(sin(\theta) - cos(\theta)\):

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

This equation provides the polar form of the original rectangular equation.

### Importance of SimplificationSimplification helps not only for clarity but also for computational purposes, especially when analyzing the behavior of polar plots. In our example, it moves from a linear Cartesian relationship to one involving division by a trigonometric function, opening new methods of analysis and visualization.