Problem 9 - Algebra and Trigonometry with Analytic Geometry

Question

\(x=2-3 \sin t, \quad y=-1-3 \cos t ; \quad 0 \leq t \leq 2 \pi\)

Step-by-Step Solution

Answer

The path is a circle centered at (2, -1) with radius 3.

Step 1: Understand the problem

We need to determine the path of a point described by the parametric equations \( x = 2 - 3 \sin t \) and \( y = -1 - 3 \cos t \) over the interval \( 0 \leq t \leq 2\pi \). These equations describe a transformation of the parametric equations of a circle.

Step 2: Recognize the Circle Form

The given parametric equations \( x = 2 - 3 \sin t \) and \( y = -1 - 3 \cos t \) can be rewritten in the form \( x = h + a \sin t \) and \( y = k + b \cos t \). This helps us recognize that the path is a circle centered at point \((h, k)\) with certain transformations applied to its standard trigonometric functions.

Step 3: Identify the Components

Recognize that \( 3 \sin t \) and \( 3 \cos t \) suggest a circle with a radius of 3. Also, from \(x = 2 - 3\sin t\), we see the center's x-coordinate is 2, and from \(y = -1 - 3\cos t\), the center's y-coordinate is -1. This means the circle is centered at \((2, -1)\).

Step 4: Analyze the Range of Values

Since \( \sin t \) and \( \cos t \) both fluctuate between -1 and 1 as \( t \) goes from 0 to \( 2\pi \), \(-3 \sin t\) and \(-3 \cos t\) will move between -3 and 3. The x-values \( x = 2 - 3 \sin t \) simplify to the range \( -1 \leq x \leq 5 \), and y-values \(-1 - 3\cos t \) to \( -4 \leq y \leq 2 \).

Step 5: Determine the Path

Based on the transformed equations \( x = 2 - 3 \sin t \) and \( y = -1 - 3 \cos t \), and knowing these are circle parametric equations, the path traced is a circle centered at \((2, -1)\) with a radius of 3.

Key Concepts

Circle Equation

A circle equation in parametric form is a convenient way to describe a circle's trajectory using parameters, typically trigonometric functions. The parametric equations resemble:

• \( x = h + a ext{cost} \)
• \( y = k + b ext{sint} \)

Here,

• \((h, k)\) is the center of the circle
• \(a\) and \(b\) are coefficients that can define a radius when they are equal

For our specific scenario, the equations \( x = 2 - 3\sin t \) and \( y = -1 - 3\cos t \) are modified parametric circle equations. The transformations indicate that the trajectory remains a circle with the center at

• \((2, -1)\)
• and a radius of 3

Trigonometric Functions

Trigonometric functions, such as \(\sin t\) and \(\cos t\), play a crucial role in creating smooth, periodic paths like circles. These functions:

• Have a repeating cycle every \(2\pi\)
• Fluctuate between -1 and 1

When these functions are multiplied by a constant, the output will scale accordingly.
In our case,

• \(-3\sin t\) ranges from -3 to 3, and shifts this sinusoidal amplitude in the x-direction
• \(-3\cos t\) affects the y-direction similarly.

These transformations form the backbone of mapping out the circular path, with each complete range of values from

• \(\sin t\) and \(\cos t\) drawing out a stable radius.

Coordinate Geometry

Coordinate geometry allows us to analyze geometric figures, like circles, through algebraic equations. In this sphere, we use coordinates described in equations:\(x = h + a\sin t\) and \(y = k + b\cos t\). The goal is to map unknown geometric forms on a plane using:

• The center of the circle, \((h, k) \)
• The range defined by \(-a\) to \(a\) and \(-b\) to \(b\)

Through such methods, we can predict the extent of paths confined by these transformations.
In our example,

• the x-values \(-1 \leq x \leq 5\)
• nice y-values \(-4 \leq y \leq 2\)
outline the boundaries within which

the circle's body lies on the cartesian plane.

Transformation of Curves

Transformation of curves deals with shifting, stretching, or rotating figures, like circles, on a coordinate plane.
The original circle equations:

• \(x = a\sin t\)
• \(y = b\cos t\)

can undergo transformations such as:

• Shifting: Moving a circle horizontally or vertically, like changing its center as in \((2, -1)\)
• Scaling: Adjusting the size of the circle through coefficients \(a\) and \(b\)

Therefore, our transformations \(x = 2 - 3\sin t\) and \(y = -1 - 3\cos t\) are simply modifying the location and extent of the circle, while the uniform radius remains intact at 3, demonstrating these transformations in action.