Problem 26 - Algebra and Trigonometry with Analytic Geometry

Question

(a) Describe the graph of a curve \(C\) that has the parametrization $$ x=-2+3 \sin t, \quad y=3-3 \cos t, \quad 0 \leq t \leq 2 \pi . $$ (b) Change the parametrization to $$ x=-2-3 \sin t, \quad y=3+3 \cos t ; \quad 0 \leq t \leq 2 \pi $$ and describe how this changes the graph from part (a). (c) Change the parametrization to $$ x=-2+3 \sin t, \quad y=3+3 \cos t ; \quad 0 \leq t \leq 2 \pi $$ and describe how this changes the graph from part (a).

Step-by-Step Solution

Answer

(a) Circle at ((-2, 3) ) with radius 3; (b) same circle, opposite traversal; (c) same circle, reflected over y-axis.

Step 1: Identify the Shape Described by Parametrization (a)

The given parametrization is: \( x = -2 + 3 \sin t \), \( y = 3 - 3 \cos t \), \( 0 \leq t \leq 2\pi \). Consider the general form \( (x-h)^2 + (y-k)^2 = r^2 \) for a circle.Rewriting the equations, we get:- \( x + 2 = 3 \sin t \) implies \( \sin t = \frac{x + 2}{3} \)- \( y - 3 = -3 \cos t \) implies \( \cos t = \frac{3 - y}{3} \)Using the identity \( \sin^2 t + \cos^2 t = 1 \), substitute to get:\[\left(\frac{x + 2}{3}\right)^2 + \left(\frac{3 - y}{3}\right)^2 = 1\]This simplifies to describe a circle centered at \((-2, 3)\) with radius 3. So, the graph is a circle centered at \((-2, 3)\) with radius 3.

Step 2: Describe the Impact of Parametrization (b)

The new parametrization is: \( x = -2 - 3 \sin t \), \( y = 3 + 3 \cos t \), \( 0 \leq t \leq 2\pi \). Instead of using the identity directly, observe how each parameter changes compared to (a).\( -2 + 3 \sin t \) changes to \(-2 - 3 \sin t\), which reflects the x-coordinate across the vertical line \(x = -2\). Similarly, \(3 - 3 \cos t\) changes to \(3 + 3 \cos t\), reflecting the y-coordinate across the horizontal line \(y = 3\).Therefore, the graph of (b) is the original circle, but traversed in the opposite direction because both parameters are negated.

Step 3: Describe the Impact of Parametrization (c)

Parametrization (c) is: \( x = -2 + 3 \sin t \), \( y = 3 + 3 \cos t \), \( 0 \leq t \leq 2\pi \). Compare this with (a):- In \( y \), \( 3 - 3 \cos t \) becomes \(3 + 3 \cos t\), indicates a reflection across the line \(y = 3\).Thus, the graph is the same circle centered at \((-2, 3)\) with radius 3, but it is reflected over the horizontal axis \(y = 3\).

Key Concepts

Circle

A circle is a simple, closed shape where all points equidistant from a center point in a plane form its boundary. Understanding when a curve represents a circle can be crucial in different areas of mathematics and geometry.
Given a general equation of circle in the form \((x - h)^2 + (y - k)^2 = r^2\), it describes a circle centered at \((h, k)\) with radius \(r\).
In this exercise, the parametrized equations \(x = -2 + 3 \sin t\) and \(y = 3 - 3 \cos t\) when transformed with trigonometric identities lead us to a recognized circular form:

• The center of the circle is \((-2, 3)\).
• The radius of the circle is \(3\).

Understanding such concepts helps mathematically graph circles accurately, and thus, interpret them correctly in various contexts.

Graph Reflections

Graph reflections involve flipping the graph of a function across a specific line, resulting in a mirror image. This is a significant concept in transformations that allows us to predict the visual outcome of altering equations or their parameters.
In part (b), we observe reflections by changing parameters:

• The change from \(-2 + 3 \sin t\) to \(-2 - 3 \sin t\) reflects the x-coordinates across the vertical line \(x = -2\).
• Similarly, changing \(3 - 3 \cos t\) to \(3 + 3 \cos t\) reflects the y-coordinates across the horizontal line \(y = 3\).

Therefore, understanding graph reflections allows us to visually and analytically comprehend how modified equations affect the position and orientation of shapes within a coordinate plane.

Trigonometric Identities

Trigonometric identities are fundamental formulas involving trigonometric functions like sine and cosine empowering us to simplify and solve complex problems.
The identity \(\sin^2 t + \cos^2 t = 1\) is specifically invaluable for determining circular forms in parametric equations.
In the exercise, parameter equations are manipulated using this identity to translate into the familiar equation of a circle. It shows us how:

• \(\left(\frac{x + 2}{3}\right)^2 + \left(\frac{3 - y}{3}\right)^2 = 1\)

Solving such identities requires not only memorization but understanding how they apply to different transformations and coordinate systems, elucidating their crucial role in mathematical problem-solving.

Parametric Curves

Parametric curves allow us to represent geometric figures in terms of parameters, typically denoted as \(t\), providing a versatile way to describe distinctive paths or motions. They provide information about each corresponding \(x\) and \(y\) as \(t\) varies over a certain interval.
The exercise shows how parametric equations can be altered to change the representation of a curve:

• The x-coordinate \(x = -2 + 3 \sin t\) and y-coordinate \(y = 3 - 3 \cos t\) define a circle.
• Modification in parameters, such as in part (b) and (c), changes reflections or orientations in graphs.

By mastering parametric curves, students can analyze complex movements and shapes, moving beyond traditional Cartesian equations and expanding their toolset for academic exploration and model depiction.