Problem 33 - College Trigonometry

Question

In Exercises \(25-39\), find a parametric description for the given oriented curve. the circle \(x^{2}+y^{2}-6 y=0\), oriented counter-clockwise

Step-by-Step Solution

Answer

The parametric equations are \(x(t) = 3 \cos(t)\) and \(y(t) = 3 + 3 \sin(t)\), for \(t \in [0, 2\pi]\).

Step 1: Rewrite the Circle Equation

The given equation is \(x^2 + y^2 - 6y = 0\). Start by completing the square for the terms involving \(y\). Rewrite the equation as \(x^2 + (y^2 - 6y) = 0\). Focus on the term \(y^2 - 6y\) to complete the square.

Step 2: Complete the Square

To complete the square for \(y^2 - 6y\), take half of the coefficient of \(y\) (which is \(-6\)), square it, and add it back into the expression. Half of \(-6\) is \(-3\), and \((-3)^2 = 9\). Rewrite the equation as \(x^2 + (y^2 - 6y + 9) = 9\). This can be further expressed as \(x^2 + (y-3)^2 = 9\).

Step 3: Identify the Circle's Characteristics

The equation \(x^2 + (y-3)^2 = 9\) represents a circle centered at \((0, 3)\) with a radius of 3. We will use this information to find parametric equations.

Step 4: Set Up Parametric Equations

For a circle oriented counter-clockwise, typical parametric equations involve trigonometric functions: \(x(t) = r \cos(t)\) and \(y(t) = r \sin(t)\). Here, \(r = 3\), and the center is \((0, 3)\), so the parametric equations will be \(x(t) = 3 \cos(t)\) and \(y(t) = 3 + 3 \sin(t)\).

Step 5: Verify the Orientation

The chosen parametric representation ensures the orientation is counter-clockwise. The function \(x(t) = 3 \cos(t)\) and \(y(t) = 3 + 3 \sin(t)\) goes from \(t = 0\) to \(t = 2\pi\) and covers the circle counter-clockwise when the standard trigonometric movement is applied.

Key Concepts

Completing the Square

The method of completing the square is a technique used to simplify quadratic expressions, especially in equations like circles or parabolas. Think of it as reorganizing a part of the expression to clearly identify the components of a squared binomial. Here's how it works:

• Take the quadratic expression in the form of \( y^2 - 6y \).
• Identify the coefficient of \( y \), which is \(-6\) in this case.
• Halve the coefficient: \(-6/2 = -3\).
• Square this result: \((-3)^2 = 9\).
• Add and subtract this square inside the original expression to maintain balance, transforming \( y^2 - 6y \) into \( (y-3)^2 \).

This transformation reveals the perfect square trinomial and helps simplify the equation into a form that's easily recognizable. In this scenario, it reveals the circle's geometric nature, making further analysis much smoother.

Circle Equation

When dealing with circles in mathematics, knowing the standard form of a circle's equation is essential. The general circle equation is \( (x - h)^2 + (y - k)^2 = r^2 \). Here:

• \((h, k)\) is the center of the circle.
• \(r\) is the radius of the circle.

By completing the square for the given equation \( x^2 + y^2 - 6y = 0 \), we rewrite it as \( x^2 + (y-3)^2 = 9 \). From this, we can easily determine:

• The center \((h, k)\) is \((0, 3)\).
• The radius \(r\) is \( \sqrt{9} = 3 \).

This explicit breakdown makes it straightforward to understand the geometric properties of the circle at hand and aids in plotting, positioning, or further transforming the circle into various forms, such as parametric.

Trigonometric Functions

Trigonometric functions are pivotal in representing periodic phenomena, like circles. For circles, the most common trigonometric functions used are sine and cosine. In parametric form, they help express the circle’s points intuitively:

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

Given our circle centered at \( (0, 3) \) with a radius of 3, the parametric equations become \( x(t) = 3 \cos(t) \) and \( y(t) = 3 + 3 \sin(t) \). The symmetry and periodic nature of sine and cosine functions seamlessly translate the geometric circle into algebraic expressions, capturing all circle points while ensuring the desired orientation.

Parametric Representation

Parametric representation of curves is a powerful mathematical tool that describes every point on a curve as a function of one or more parameters. For circles, this method elegantly expresses circular paths using trigonometric functions:

• The parameter \(t\) often represents the angle from the positive x-axis, moving counter-clockwise.
• Equations are structured as \( x(t) \) and \( y(t) \), which in our example, are derived as \( x(t) = 3 \cos(t) \) and \( y(t) = 3 + 3 \sin(t) \).

These equations provide a clear path for tracing the circle, ensuring completeness and precision in representation. Parametric equations are not only crucial for circles but extend to a variety of other curves, allowing for pinpoint placement and movement along any curved path. They are clearer than traditional equations when movement direction or exact plot is required.