Problem 23 - Calculus Early Transcendentals

Question

\(21-26\) Find a polar equation for the curve represented by the given Cartesian equation. $$x=-y^{2}$$

Step-by-Step Solution

Answer

The polar equation is \( \cos \theta + r \sin^2 \theta = 0 \).

Step 1: Understand Cartesian to Polar Conversion

The given Cartesian equation is \( x = -y^2 \). To convert a Cartesian equation to a polar equation, recall the relationships between polar and Cartesian coordinates: \( x = r \cos \theta \) and \( y = r \sin \theta \).

Step 2: Substitute Polar Equations

Substitute the polar coordinates into the given Cartesian equation: Replace \( x \) with \( r \cos \theta \) and \( y \) with \( r \sin \theta \). The equation becomes \( r \cos \theta = -(r \sin \theta)^2 \).

Step 3: Simplify the Equation

Simplify the equation derived in the previous step: \( r \cos \theta = -r^2 \sin^2 \theta \). This equation can be rewritten as \( r \cos \theta + r^2 \sin^2 \theta = 0 \).

Step 4: Factor the Equation

Factor out the common term \( r \): \( r(\cos \theta + r \sin^2 \theta) = 0 \).

Step 5: Analyze and Solve

For the equation \( r(\cos \theta + r \sin^2 \theta) = 0 \), there are two possible solutions: either \( r = 0 \) or \( \cos \theta + r \sin^2 \theta = 0 \). The solution \( r = 0 \) represents the origin, while \( \cos \theta + r \sin^2 \theta = 0 \) is the desired polar equation.

Key Concepts

Cartesian to Polar Conversion

When you're faced with a Cartesian equation like \( x = -y^2 \), converting it to polar coordinates can lead to interesting insights. The conversion process uses two key relationships between the Cartesian coordinates (\(x, y\)) and their polar counterparts (\(r, \theta\)). The formulas you'll need are:

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

These relationships help translate any point or equation from the familiar XY-plane (Cartesian) to the circular motion around a point (polar). Once you've substituted these expressions into your equation, you'll be able to express the whole equation in terms of \(r\) and \(\theta\), which can sometimes make it simpler or reveal more about the nature of the curve.

Coordinate Relationships

Coordinate relationships are fundamental to converting and understanding mathematical equations in different forms. Knowing how the Cartesian coordinates relate to polar coordinates allows for seamless translation between these two systems.
For instance, the expressions \(x = r \cos \theta\) and \(y = r \sin \theta\) show how Cartesian coordinates link to polar coordinates.
These equations imply that you multiply the radius \(r\) by the cosine or sine of an angle \(\theta\) to find the positional components \(x\) and \(y\). This becomes incredibly useful when transforming curves and analyzing their properties in a circular context.

Equation Simplification

Simplifying equations is a critical step in math to make them easier to handle, interpret, and solve. In our example, once the Cartesian equation \( x = -y^2 \) is converted to polar form, you get \( r \cos \theta = -(r \sin \theta)^2 \).
To simplify:

• Combine terms, rewriting it as \( r \cos \theta = -r^2 \sin^2 \theta \).
• Add \(r^2 \sin^2 \theta \) to both sides to obtain \( r \cos \theta + r^2 \sin^2 \theta = 0 \).

This form often reveals more about the structure of the equation, making it easier to identify solutions or specific behavior.
Simplification doesn't just make equations look nicer; it often highlights underlying patterns or solutions that aren't obvious at first glance.

Factorization

Factorization in mathematics involves rewriting an equation or expression as a product of simpler terms. It’s a powerful tool because it can reveal solutions that are hidden in a more complicated form.
In the case of the equation \( r \cos \theta + r^2 \sin^2 \theta = 0 \), factorization allows us to take out \( r \) as a common factor:

• The equation becomes \( r(\cos \theta + r \sin^2 \theta) = 0 \).

This form immediately shows that the value of \(r\) can either be zero (implying the origin in polar terms) or that the expression inside the parentheses must be zero.
This technique helps in breaking down complex problems into manageable chunks and finding simple conditions for solutions.

Polar Coordinates

Polar coordinates provide a unique way of identifying a location based on angle and distance from a fixed point, usually the origin.
Instead of relying on horizontal and vertical distances (as with Cartesian coordinates), polar coordinates use a radius \(r\) and an angle \(\theta\).
This approach is particularly useful for circular and rotational phenomena.

• In equations, polar coordinates can simplify expressions with symmetry or periodicity.
• For the equation \( x = -y^2 \) translated into polar form, we see how the relationship changes variables but potentially simplifies the understanding of the curve.

Ultimately, polar coordinates offer potent insight into different mathematical contexts and are frequently used in fields like physics, engineering, and geography.