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Magazine Chapter 11: Problem 9
\(x=2-3 \sin t, \quad y=-1-3 \cos t ; \quad 0 \leq t \leq 2 \pi\)
Short Answer
Expert verified
The path is a circle centered at (2, -1) with radius 3.
Step by step solution
01
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.
02
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.
03
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 \((2, -1)\).
04
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 \( -4 \leq y \leq 2 \).
05
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.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
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:
- \( x = h + a ext{cost} \)
- \( y = k + b ext{sint} \)
- \((h, k)\) is the center of the circle
- \(a\) and \(b\) are coefficients that can define a radius when they are equal
- \((2, -1)\)
- 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:
In our case,
- Have a repeating cycle every \(2\pi\)
- Fluctuate between -1 and 1
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.
- \(\sin t\) 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:
In our example,
- The center of the circle, \((h, k) \)
- The range defined by \(-a\) to \(a\) and \(-b\) to \(b\)
In our example,
- the x-values \(-1 \leq x \leq 5\)
- nice y-values \(-4 \leq y \leq 2\) outline the boundaries within which
Transformation of Curves
Transformation of curves deals with shifting, stretching, or rotating figures, like circles, on a coordinate plane.
The original circle equations:
The original circle equations:
- \(x = a\sin t\)
- \(y = b\cos t\)
- 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 \(b\)
