Problem 71
Question
Plot the following parametric curves. Describe in words how the point moves around the curve in each case. (a) \(x=\cos \left(t^{2}-t\right), y=\sin \left(t^{2}-t\right)\) (b) \(x=\cos \left(2 t^{2}+3 t+1\right), y=\sin \left(2 t^{2}+3 t+1\right)\) (c) \(x=\cos (-2 \ln t), y=\sin (-2 \ln t)\) (d) \(x=\cos (\sin t), y=\sin (\sin t)\)
Step-by-Step Solution
Verified Answer
Each curve represents rotation around a unit circle or an oscillation due to the sine and cosine composition.
1Step 1: Set Up the Parametric Equations
Identify the given equations for each sub-question.(a) For part (a), the equations are \(x = \cos(t^2 - t)\) and \(y = \sin(t^2 - t)\).(b) For part (b), the equations are \(x = \cos(2t^2 + 3t + 1)\) and \(y = \sin(2t^2 + 3t + 1)\).(c) For part (c), the equations are \(x = \cos(-2 \ln t)\) and \(y = \sin(-2 \ln t)\).(d) For part (d), the equations are \(x = \cos(\sin t)\) and \(y = \sin(\sin t)\).
2Step 2: Determine the Shape of Each Curve
For each part, realize that both \(x\) and \(y\) are expressed as cosines and sines of the same expression. This implies each curve is a motion along a unit circle in the polar coordinate system where the angle is given by the expression inside the cosine and sine functions.
3Step 3: Analyze the Point Movement for Each Curve
(a) As \(t\) increases, \(t^2 - t\) initially grows, then accelerates since \(t^2\) dominates, causing counterclockwise movement tracing out the unit circle repetitively due to the periodic nature of sine and cosine.(b) The term \(2t^2 + 3t + 1\) increases more rapidly due to the \(t^2\) term, again causing counterclockwise motion at an increasing angular rate, creating tighter spirals when plotted.(c) The parameter \(-2 \ln t\) moves from positively increasing to differentiating in direction depending on \(t\). This causes the curve to spiral inward/outward uniquely among \(t\)'s defined range.(d) Here \(\sin t\) varies between -1 and 1, creating simple oscillation back and forth along a small segment of the unit circle between angles \(-\frac{\pi}{2}\) and \(\frac{\pi}{2}\).
4Step 4: Plot the Parametric Curves
Using graphing software or graph paper, plot each set of equations using \(t\) as an interval, typically from a small positive number to larger values like 10, to observe how the point moves around the circle for the given \(x\) and \(y\). For example:(a) Vary \(t\) from 0 to 5 in small increments.(b) Choose \(t\) from -2 to 2 for broader observation.(c) Since the logarithm requires \(t > 0\), vary \(t\) from 0.1 upwards.(d) Examine \(t\) between 0 and \(2\pi\) for one full oscillation review of sine effect.
Key Concepts
Unit CirclePolar Coordinate SystemTrigonometric FunctionsGraphing Techniques
Unit Circle
The unit circle is a fundamental concept used often in mathematics, especially in trigonometry. It is a circle with a radius of 1, centered at the origin of a coordinate plane. The unit circle is very useful for understanding the behavior of trigonometric functions.
On the unit circle, each angle corresponds to a unique point, which helps us visualize sine and cosine values. For any point on the unit circle, the x-coordinate is the cosine of the angle, and the y-coordinate is the sine. This means any point can be represented as \(x = \cos(\theta), y = \sin(\theta)\).
Parametric curves described by trigonometric functions often lie on the unit circle, which simplifies their analysis. In the original exercise, this concept helps us understand the movement described by the parametric equations.
On the unit circle, each angle corresponds to a unique point, which helps us visualize sine and cosine values. For any point on the unit circle, the x-coordinate is the cosine of the angle, and the y-coordinate is the sine. This means any point can be represented as \(x = \cos(\theta), y = \sin(\theta)\).
Parametric curves described by trigonometric functions often lie on the unit circle, which simplifies their analysis. In the original exercise, this concept helps us understand the movement described by the parametric equations.
Polar Coordinate System
The polar coordinate system is another way to represent points on a plane, especially useful for dealing with circular and spiral patterns. Instead of using x and y coordinates, it uses an angle and a distance from a central point.
In polar coordinates, every point on a plane is described by an angle \(\theta\) and a radius \(r\). The relationship between polar and Cartesian (x, y) coordinates is described by:
The parameters given in the original exercise, such as \(t^2 - t\) or \(\sin t\), basically serve as the angle \(\theta\) in the polar coordinate system. This results in plots that often mimic circular paths or spirals around the unit circle.
In polar coordinates, every point on a plane is described by an angle \(\theta\) and a radius \(r\). The relationship between polar and Cartesian (x, y) coordinates is described by:
- \(x = r \cos(\theta)\)
- \(y = r \sin(\theta)\)
The parameters given in the original exercise, such as \(t^2 - t\) or \(\sin t\), basically serve as the angle \(\theta\) in the polar coordinate system. This results in plots that often mimic circular paths or spirals around the unit circle.
Trigonometric Functions
Trigonometric functions, namely sine and cosine, are fundamental in analyzing periodic and oscillatory phenomena. These functions describe the x and y coordinates of points on the unit circle based on a given angle.
The sine function, \(\sin(\theta)\), describes how the y-coordinate of a point changes as you move around the circle. Similarly, \(\cos(\theta)\) describes the x-coordinate. These functions are periodic with a period of \(2\pi\), meaning they repeat every \(2\pi\) radians.
In the context of parametric curves, these functions define complex shapes and paths, as they dictate how a point revolves around the unit circle. In parametric equations, the angle \(\theta\) varies with time or another parameter, leading to a dynamic representation of curves.
The sine function, \(\sin(\theta)\), describes how the y-coordinate of a point changes as you move around the circle. Similarly, \(\cos(\theta)\) describes the x-coordinate. These functions are periodic with a period of \(2\pi\), meaning they repeat every \(2\pi\) radians.
In the context of parametric curves, these functions define complex shapes and paths, as they dictate how a point revolves around the unit circle. In parametric equations, the angle \(\theta\) varies with time or another parameter, leading to a dynamic representation of curves.
Graphing Techniques
Graphing techniques are essential tools for visualizing mathematical concepts, especially for understanding parametric curves and their behavior. By plotting parametric equations, you can see how points move over time and describe complex motion patterns.
To graph parametric curves, you need to consider:
To graph parametric curves, you need to consider:
- The range of the parameter \(t\)
- The increments of \(t\)
- The interpretation of plotted points
Other exercises in this chapter
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