Problem 25 - Calculus

Question

Substitute \(r=\sin \theta\) into \(x=r \cos \theta\) and \(y=r \sin \theta\) to find \(x\) and \(y\) in terms of the parameter \(\theta .\) Then compute \(x^{2}+y^{2}\) to reach the \(x y\) cquation.

Step-by-Step Solution

Answer

The equation is \(x^2 + y^2 = y\).

Step 1: Substitute for x

We need to replace the expression for \(r\) in terms of \(\theta\) as given, which is \(r = \sin \theta\). Substitute this into \(x = r \cos \theta\):\[x = \sin \theta \cdot \cos \theta\]

Step 2: Substitute for y

Similarly, substitute \(r = \sin \theta\) into the equation for \(y\), which is \(y = r \sin \theta\):\[y = \sin \theta \cdot \sin \theta = \sin^2 \theta\]

Step 3: Calculate x² + y²

Now, calculate \(x^2 + y^2\) using the expressions derived:\[x^2 = (\sin \theta \cos \theta)^2 = \sin^2 \theta \cos^2 \theta\]\[y^2 = (\sin^2 \theta)^2 = \sin^4 \theta\]Add \(x^2\) and \(y^2\) together:\[x^2 + y^2 = \sin^2 \theta \cos^2 \theta + \sin^4 \theta\]

Step 4: Tidy the expression

Notice we can factor \(\sin^2 \theta\) from \(\sin^2 \theta \cos^2 \theta + \sin^4 \theta\), so:\[x^2 + y^2 = \sin^2 \theta (\cos^2 \theta + \sin^2 \theta)\]Since \(\cos^2 \theta + \sin^2 \theta = 1\), this simplifies to:\[x^2 + y^2 = \sin^2 \theta\]

Step 5: Final Equation Form

Since \(x = \sin \theta \cos \theta\) and \(y = \sin^2 \theta\), and now we know \(x^2 + y^2 = \sin^2 \theta\), our \(xy\) equation is:\[x^2 + y^2 = y\]

Key Concepts

Parametric Equations

Parametric equations are a way to express geometric shapes using parameters, often denoted as \( t \) or \( \theta \) in trigonometric contexts, to define the coordinates \( x \) and \( y \) of points on a curve. Instead of defining \( y \) explicitly as a function of \( x \), parametric equations use a third variable to describe the relation between variables.

• In our example, \( x \) and \( y \) are expressed in terms of \( \theta \), with \( r = \sin \theta \).
• This parameterization is useful especially in cases involving curves that are not functions, like circles or spirals.

By substituting \( r = \sin \theta \) into the standard parametric equations for polar coordinates \( x = r \cos \theta \) and \( y = r \sin \theta \), we can express \( x \) and \( y \) in terms of a single parameter \( \theta \). This transformation allows us to unravel complex shapes into manageable equations that are easier to analyze and manipulate.

Trigonometric Substitution

Trigonometric substitution is a technique that uses trigonometric identities to simplify expressions, particularly useful when dealing with integrals or equations involving roots, squares, and sum of squares. This substitution often involves recognizing patterns and using identities like \( \sin^2 \theta + \cos^2 \theta = 1 \) to reduce complexity.
For our given problem:

• We utilize the substitution \( r = \sin \theta \). This decision leverages properties of trigonometric identities to express \( x = r \cos \theta = \sin \theta \cos \theta \).
• For \( y = \sin^2 \theta \), it's a straightforward substitution given the polar coordinate transformations.

By applying these trigonometric identities, calculations become simpler, allowing us to exploit well-known trigonometric identities. This often leads to surprises, like finding our problem's essential expression \( x^2 + y^2 = \sin^2 \theta \) simplifies to \( x^2 + y^2 = y \) through identity application.

Equation Simplification

Equation simplification is the process of refining algebraic expressions to make them easier to work with. It involves applying algebraic and trigonometric rules to reduce expressions while preserving equality.

• In this problem, simplify each step by substituting \( r \) in terms of \( \theta \) and evaluating \( x^2 + y^2 \).
• Once expressions for \( x \) and \( y \) are derived, compute \( x^2 + y^2 \), factor out common terms, and apply identities such as \( \sin^2 \theta + \cos^2 \theta = 1 \).

The simplification process not only tidies up equations but reveals important results, such as associating \( x^2 + y^2 \) directly to \( y \). This connection can sometimes expose underlying geometry or symmetries, in this case confirming the nature of the parametric expressions.