Problem 19 - Precalculus

Question

Convert to polar form. $$y^{2}=x^{3}$$

Step-by-Step Solution

Answer

The polar form is \( r = \frac{\sin^2 \theta}{\cos^3 \theta} \).

Step 1: Identify Variables for Conversion

To convert the equation to polar form, identify the variables: \( x \) and \( y \). In polar coordinates, these correspond to \( r \cos \theta \) for \( x \) and \( r \sin \theta \) for \( y \).

Step 2: Substitute Polar Equivalents

Substitute \( x = r \cos \theta \) and \( y = r \sin \theta \) into the given equation \( y^2 = x^3 \). This yields: \[ (r \sin \theta)^2 = (r \cos \theta)^3 \]

Step 3: Simplify the Equation

Simplify the equation: \[ r^2 \sin^2 \theta = r^3 \cos^3 \theta \]Divide both sides by \( r^2 \) (assuming \( r eq 0 \)) to get: \[ \sin^2 \theta = r \cos^3 \theta \]

Step 4: Solve for \( r \)

Solve the equation \( \sin^2 \theta = r \cos^3 \theta \) for \( r \): \[ r = \frac{\sin^2 \theta}{\cos^3 \theta} \]

Key Concepts

Coordinate Conversion

Coordinate conversion plays a crucial role when transitioning from Cartesian (also known as rectangular) coordinates to polar coordinates. This process involves changing from the standard x and y coordinates of a point to the polar coordinates, which are represented by two values: the distance from the origin, often denoted as \( r \), and the angle \( \theta \), which is the counterclockwise angle measured from the positive x-axis.
The given exercise starts with an equation in the Cartesian coordinate system and transforms it into polar form. The task requires recognizing how x and y relate to \( r \) and \( \theta \):

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

By substituting these relationships into the initial Cartesian equation, you initiate the conversion process. This mathematical maneuver allows the representation of the same curve, but in terms of its position relative to a central point and direction rather than along a grid. Mastering this conversion helps in analyzing and interpreting curves and shapes in a different coordinate system.

Trigonometry

Trigonometry is essential in the process of converting Cartesian equations to their polar forms. In this context, trigonometric functions serve as the bridge linking two coordinate systems. These functions help express the relationship between linear positions (x and y) and angular positions (\( r \) and \( \theta \)).
Substituting into trigonometric terms is straightforward:

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

These expressions are used to rewrite the equation of a curve using polar coordinates effortlessly. For instance, using \( y = r \sin \theta \) and \( x = r \cos \theta \), the original equation \( y^2 = x^3 \) becomes an equation involving \( r \) and \( \theta \). With trigonometry, one can reconcile the different representations of geometric phenomena seamlessly.

Precalculus Mathematics

Precalculus mathematics involves a skillful understanding of various mathematical properties, including functions, geometry, and coordinate systems. This knowledge base is pivotal in handling problems like coordinate conversions. In the example problem, the simplification of the equation derived from substituting polar equivalents is an application of precalculus mathematics.
Simplification here involves dealing with terms like \( r^2 \sin^2 \theta \) and \( r^3 \cos^3 \theta \). It's essential to know:

• How to manipulate algebraic expressions consistently
• The rules for dividing expressions by using properties of exponents (as seen where \( r^2 \) is divided out)

This exercise also demonstrates derivative skills that precalculus students heavily rely upon, creating a foundational stepping stone towards calculus. Having a solid grounding in precalculus mathematics enables students to confidently approach problem-solving that involves transitioning between different mathematical domains.