Problem 64 - Algebra and Trigonometry

Question

Convert the polar equation to rectangular coordinates. $$ r=\frac{2}{1-\cos \theta} $$

Step-by-Step Solution

Answer

The rectangular form is \( y^2 = 4x + 4 \).

Step 1: Identify Polar Equation Components

The given polar equation is \( r = \frac{2}{1 - \cos \theta} \). We recognize the components: \( r \) is the radial distance, and \( \theta \) is the angle.

Step 2: Express \( \cos \theta \) in Rectangular Coordinates

We use the identity \( \cos \theta = \frac{x}{r} \) to express \( \cos \theta \) in terms of \( x \) and \( r \).

Step 3: Substitute \( \cos \theta \) into the Equation

Replace \( \cos \theta \) in the equation with \( \frac{x}{r} \). The equation becomes: \[ r = \frac{2}{1 - \frac{x}{r}} \].

Step 4: Simplify and Clear the Fraction

To eliminate the fraction, multiply both sides by \( 1 - \frac{x}{r} \), resulting in: \[ r(1 - \frac{x}{r}) = 2 \]. Simplify this to \( r - x = 2 \).

Step 5: Substitute \( r \) with its Rectangular Form

To convert from polar to rectangular coordinates, substitute \( r \) with \( \sqrt{x^2 + y^2} \). The equation becomes: \[ \sqrt{x^2 + y^2} - x = 2 \].

Step 6: Solve for a Rectangular Form

Rearrange \( \sqrt{x^2 + y^2} = x + 2 \) and square both sides to remove the square root: \[ x^2 + y^2 = (x + 2)^2 \].

Step 7: Expand and Simplify

Expand \( (x + 2)^2 \) to get \( x^2 + 4x + 4 \). Substitute this back into \( x^2 + y^2 = x^2 + 4x + 4 \) to obtain \( y^2 = 4x + 4 \).

Step 8: Final Rectangular Equation

Therefore, the rectangular form of the equation is: \[ y^2 = 4x + 4 \].

Key Concepts

Polar Coordinates

Polar coordinates are a way of representing points in a plane using two values: the radial distance from the origin and the angle from the positive x-axis. Rather than using x and y like rectangular (Cartesian) coordinates, polar coordinates use \( r \) for the distance and \( \theta \) for the angle. This system is especially useful in cases involving circular or spiral patterns, as they naturally fit these shapes.
Understanding the conversion between polar and rectangular coordinates is crucial. A key to this is recognizing how the polar coordinates correspond to rectangular coordinates, where \( x = r \cos \theta \) and \( y = r \sin \theta \). This relation allows you to move between the two coordinate systems efficiently. It’s essential to become adept at these transformations, as they're widely used in physics, engineering, and mathematics.

Trigonometric Identities

Trigonometric identities are formulas that express relationships between the angles and sides of triangles. They are fundamental in math because they allow the conversion between various trigonometric functions. A crucial identity used in coordinate conversion is \( \cos \theta = \frac{x}{r} \), which transforms the angle \( \theta \) in terms of a known \( x \) and distance \( r \).
These identities also help simplify polar equations when converting into rectangular form. Understanding and memorizing basic trigonometric identities, like \( \sin^2 \theta + \cos^2 \theta = 1 \), is invaluable as they frequently appear in math problems. These tools reduce complex expressions and facilitate solving equations involving more complicated trigonometric elements.

Coordinate Conversion

Coordinate conversion involves translating a point or equation from one coordinate system, like polar, to another, such as rectangular. This technique is necessary when a problem is more easily solved or understood in a different coordinate system.
To perform this conversion, you apply relationships between polar and rectangular coordinates. For example, given a point in polar form \((r, \theta)\), it can be rewritten as \( x = r \cos \theta \) and \( y = r \sin \theta \) in rectangular form. In the given problem, we converted the equation \( r = \frac{2}{1 - \cos \theta} \) into rectangular coordinates by replacing components using these transformations. This method requires careful substitution steps and algebraic manipulation to ensure accuracy.

Equation Conversion

Equation conversion is the process of translating equations from one form to another, often to make them more tractable or aesthetically pleasing. When you convert a polar equation to a rectangular one, you resolve the equation using trigonometric relationships and algebra.
In our exercise, we began with the polar equation \( r = \frac{2}{1 - \cos \theta} \) and used the identity \( \cos \theta = \frac{x}{r} \). Through substituting this identity and subsequent algebraic simplification, we resulted in the rectangular equation \( y^2 = 4x + 4 \).
Such conversion often involves clearing fractions, distributing terms, and sometimes, squaring sides to resolve square roots. Mastery in manipulating equations between forms is a vital skill for any math student, enhancing both problem-solving capabilities and understanding of mathematical relationships.