Problem 55 - Algebra and Trigonometry

Question

Convert each rectangular equation to a polar equation that expresses r in terms of \(\theta\). $$ (x-2)^{2}+y^{2}=4 $$

Step-by-Step Solution

Answer

The polar equation corresponding to the given rectangular equation is \(r = 4\cos\theta\).

Step 1: Replace x and y with their polar equivalents

Begin by substituting \(x = r \cos\theta\) and \(y = r \sin\theta\) into the original equation. This results in: \((r\cos\theta - 2)^{2}+(r\sin\theta)^{2}=4\)

Step 2: Expand the equation and simplify

Expand and simplify the resulting equation, combine like terms and equate it to \(r^2\): \(r^2 \cos^2\theta - 4r\cos\theta + 4 + r^2 \sin^2\theta = 4 => r^2 (\cos^2\theta + \sin^2\theta) - 4r\cos\theta = 0 .\) Now, knowing \(\cos^2\theta + \sin^2\theta = 1\), the simplification results in: \(r^2 - 4r\cos\theta = 0 \).

Step 3: Solve for r

Now, solve the simplified equation for \(r\), by factoring out \(r\): \(r(r - 4\cos\theta) = 0\). Therefore, r is: \(r = 0\) or \(r = 4\cos\theta\)

Key Concepts

Rectangular to Polar Conversion

When converting a rectangular equation into a polar equation, it's essential to familiarize yourself with the relationship between rectangular coordinates \(x, y\) and polar coordinates \(r, \theta\). In polar coordinates, any point can be described using the distance from the origin, represented by \(r\), and the angle \(\theta\), which is measured from the positive x-axis.
To transition from rectangular to polar equations, we use the following transformation formulas:

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

By substituting these into the given rectangular equation, you can convert it into a polar form. This is demonstrated with the exercise, where we replaced \(x\) with \(r \cos\theta\) and \(y\) with \(r \sin\theta\), bringing us closer to the desired polar equation.

Trigonometric Identities

Trigonometric identities are crucial tools in mathematics, especially when simplifying equations that involve trigonometric expressions. One of the most frequently used identities for transformations between polar and rectangular forms is \(\cos^2\theta + \sin^2\theta = 1\).
This identity simplifies the total squared trigonometric components when they are on either side of an equation. In the solution above, after expanding the polar expression, we apply \(\cos^2\theta + \sin^2\theta = 1\) to combine the terms \(r^2 \cos^2\theta + r^2 \sin^2\theta\) into \(r^2\). This is crucial for simplifying the complicated expressions that occur after converting the coordinates, allowing for the creation of simpler and easily solvable polar equations.

Equation Simplification

Equation simplification requires combining like terms and solving for one variable. In this case, after applying our trigonometric identities, we arrive at an equation \(r^2 - 4r\cos\theta = 0\).
This equation was simplified by recognizing a common factor, \(r\), which is then factored out, leaving us with two potential solutions: \(r = 0\) or \(r = 4\cos\theta\).
By simplifying the equation in this manner, not only do we find possible solutions, but it also makes the equation easier to work with in subsequent mathematical processes or applications. Understanding how to efficiently simplify is instrumental, as it turns the challenge of multiple terms into a straightforward solution.