Problem 106 - Precalculus with Limits

Question

In Exercises 85-108, convert the polar equation to rectangular form. \(r=\dfrac{1}{1-\cos\ \theta}\)

Step-by-Step Solution

Answer

The rectangular form of the given polar equation is \(x^2+y^2-x-r=0\).

Step 1 Convert r and θ to rectangular coordinates

To convert the polar equation \(r=\dfrac{1}{1-\cos\ \theta}\) into rectangular form, we first note that in rectangular coordinates, \(r^2=x^2+y^2\) and \(\cos \theta = \frac{x}{r}\). Substitute \(\cos \theta\) with \(\frac{x}{r}\) in the polar equation.

Step 2 Simplification

Substituting \(\cos \theta\) with \(\frac{x}{r}\) will yield the equation \(r=\dfrac{1}{1-\frac{x}{r}}\). Multiplying both sides of the equation by \((1-\frac{x}{r})\) we get \(r-\frac{x}{r^2} = 1\). Now multiply the entire equation by r to get rid of the denominator. This will give us \(r^2-x= r\). We can replace \(r^2\) with \(x^2+y^2\) on the left side.

Step 3 Final Answer

Our rectangular form of \(r=\dfrac{1}{1-\cos\ \theta}\) is \(x^2+y^2-x = r \) after replacing \(r^2\) with \(x^2+y^2\). In other words we have \(x^2+y^2-x - r = 0\).

Key Concepts

Polar Coordinates

Polar coordinates are a system used to describe the location of a point in a plane by specifying its distance from a fixed point, known as the origin, and the angle formed with a fixed direction, usually the positive x-axis. This system is especially useful in scenarios where understanding relationships between lengths and angles is crucial, such as navigation and astronomy.
In polar coordinates, a point P is represented by the pair \(r, \theta\), where:

• \(r\) is the radial coordinate, indicating the distance from the origin to the point P.
• \(\theta\) is the angular coordinate, representing the counterclockwise angle from the positive x-axis to the line segment connecting the origin to point P.

These coordinates are advantageous in problems involving symmetry or radial patterns, as they simplify the description of circular paths and formations. Converting expressions from polar to rectangular coordinates can often provide useful analytical geometric insights.

Rectangular Coordinates

Rectangular coordinates, also known as Cartesian coordinates, define a point in the plane using two perpendicular lines, typically called the x-axis and y-axis. Each point is represented as \(x, y\), where:

• \(x\) is the horizontal component, indicating the point's position along the x-axis.
• \(y\) is the vertical component, representing the position along the y-axis.

This coordinate system is ideal for graphing linear equations and performing vector calculations. It breaks down problems into simple, manageable components, making it easier to analyze straight line distances and intersections.

When converting from polar to rectangular coordinates, the relationships between \(x, y\), \(r\), and \(\theta\) are essential:

• \(x = r \cos\theta\)
• \(y = r \sin\theta\)
• \(r = \sqrt{x^2 + y^2}\)
• \(\theta = \tan^{-1}(\frac{y}{x})\)

These transformations help bridge the gap between the polar and rectangular descriptions of a point in the plane.

Trigonometric Substitution

Trigonometric substitution is a technique used to simplify the integration or transformation of expressions involving trigonometric functions. In the realm of coordinate conversion, it helps us handle the trigonometric components efficiently. In the context of converting polar equations to rectangular form, trigonometric identities provide the necessary substitutions.

Consider the polar equation given: \(r = \frac{1}{1 - \cos\theta}\). Substitution involves:

• Recognizing that \(\cos\theta = \frac{x}{r}\) in rectangular terms.
• Replacing \(\cos\theta\) using this relationship to transform the polar equation.
• Handling the resulting equation to express it in terms of \(x\) and \(y\).

These identities and substitutions help bridge the two systems, ensuring continuity in analysis across different mathematical contexts. By strategically applying these substitutions, complex geometrical problems can become much more tractable and easier to handle in any chosen coordinate system.