Problem 64 - Calculus Early Transcendentals

Question

Convert the following equations to polar coordinates. \((x-1)^{2}+y^{2}=1\)

Step-by-Step Solution

Answer

Question: Convert the Cartesian equation \((x-1)^{2}+y^{2}=1\) into polar coordinates. Answer: The Cartesian equation \((x-1)^{2}+y^{2}=1\) is equivalent to the polar equation \(r(r-2\cos\theta) = 0\).

Step 1: Substitute polar coordinate conversion formulas

Replace \(x\) with \(r\cos\theta\) and \(y\) with \(r\sin\theta\) in the given equation: \([(r\cos\theta)-1]^{2}+(r\sin\theta)^2 = 1\)

Step 2: Expand and Simplify

First, we expand the square terms: \((r^2\cos^2\theta - 2r\cos\theta + 1) + r^2\sin^2\theta = 1\) Now, combine the \(r^2\cos^2\theta\) and \(r^2\sin^2\theta\) terms using the identity \(\cos^2\theta + \sin^2\theta = 1\): \(r^2(\cos^2\theta + \sin^2\theta) - 2r\cos\theta + 1 = 1\) And replace \(\cos^2\theta + \sin^2\theta\) with 1: \(r^2 - 2r\cos\theta + 1 = 1\)

Step 3: Solve for Polar Equation

Subtract 1 from both sides of the equation: \(r^2 - 2r\cos\theta = 0\) Factor out \(r\) from the equation: \(r(r - 2\cos\theta) = 0\) Now, we have the polar equation: \(r(r - 2\cos\theta) = 0\) The given Cartesian equation \((x-1)^{2}+y^{2}=1\) is equivalent to the polar equation \(r(r-2\cos\theta) = 0\).

Key Concepts

Trigonometric Identities

Trigonometric identities are essential tools when working with different coordinate systems, especially in polar coordinates conversion. One such identity used frequently is the Pythagorean identity. This identity states that for any angle \(\theta\), the sum of the squares of sine and cosine is equal to one: \[ \cos^2\theta + \sin^2\theta = 1 \]This identity helps simplify expressions involving sine and cosine. When converting Cartesian equations to polar form, as shown in the step-by-step solution, we used this identity to combine terms like \(r^2\cos^2\theta\) and \(r^2\sin^2\theta\) to simplify them to \(r^2\). Understanding and applying these identities can make equation simplification much more manageable, especially when dealing with trigonometric functions.

Coordinate Systems

Coordinate systems provide a framework for locating points in a plane. The two most common systems are Cartesian (rectangular) and Polar.

• The Cartesian system uses \(x\) and \(y\) coordinates based on horizontal and vertical lines.
• The Polar system, however, uses \(r\) (the radius or distance from the origin) and \(\theta\) (the angle from the positive x-axis).

When converting from Cartesian to Polar coordinates, we substitute \(x = r \cos \theta\) and \(y = r \sin \theta\). This conversion allows us to express points in terms of distance and angle, which can be particularly useful in scenarios involving rotations or circular paths.In our exercise, converting the Cartesian equation \((x-1)^2 + y^2 = 1\) into polar form illustrates how Polar coordinates transform the way equations are represented, emphasizing the relationship between distance and angle.

Algebraic Manipulation

Algebraic manipulation involves rearranging and simplifying expressions to solve equations effectively. In the context of converting equations to polar coordinates, it involves expanding terms, using identities, and reducing equations to simpler forms.For the equation \([(r\cos\theta)-1]^{2}+(r\sin\theta)^2 = 1\), we start by expanding the squared terms. This type of manipulation is crucial for isolating terms, making it easier to apply trigonometric identities. The goal in algebraic manipulation is to rewrite equations in a form where patterns or identities can be recognized and applied. Once simplified, the expression \(r^2 - 2r\cos\theta + 1 = 1\) was achieved by first expanding, then using the identity \(\cos^2\theta + \sin^2\theta = 1\). From this point, solving becomes more straightforward by factoring out common terms, leading us to the final polar equation.Algebraic manipulation is a powerful tool that often involves combining simplification strategies with known identities to yield clear and efficient solutions.