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x = sin´t, y = cosʻt, 0

x = sin´t, y = cosʻt, 0

Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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Find the distance traveled by a particle with position sx, yd as t varies in the given time interval. Compare with the length of the curve.

**Problem 55.** Given the parametric equations: - \( x = \sin^2 t \) - \( y = \cos^2 t \) with the parameter \( t \) in the range \( 0 \leq t \leq 3\pi \). These equations represent a transformation of the trigonometric functions sine and cosine, squared, to define a curve or a relation between \( x \) and \( y \) over the specified interval of \( t \). ### Explanation of Graphical Representation If one were to graph these equations in an \( xy \)-coordinate system: 1. **Initial Observation**: - As \( t \) increases from 0 to \( \pi/2 \), \( x \) increases from 0 to 1 and \( y \) decreases from 1 to 0. - From \( \pi/2 \) to \( \pi \), \( x \) decreases from 1 to 0 and \( y \) increases from 0 to 1. - This behavior repeats in each interval of length \( \pi \). 2. **Graph Shape**: - The path described is a parabolic segment or a sinusoidal variation between two mirrored curves, owing to the equations involving squares of sine and cosine. - The entire path is symmetric across the lines \( x = y \) and \( y = x \). 3. **Repetitions**: - Given \( t \) runs from 0 to \( 3\pi \), this behavior repeats three times, cycling through from a start at the origin through peak points at \( (1, 0) \) and back to another symmetric point at \( (0, 1) \). Understanding these parametric equations through their graphical interpretation and algebraic manipulation can help in analyzing trigonometric behavior and the nature of such periodic functions.
Transcribed Image Text:**Problem 55.** Given the parametric equations: - \( x = \sin^2 t \) - \( y = \cos^2 t \) with the parameter \( t \) in the range \( 0 \leq t \leq 3\pi \). These equations represent a transformation of the trigonometric functions sine and cosine, squared, to define a curve or a relation between \( x \) and \( y \) over the specified interval of \( t \). ### Explanation of Graphical Representation If one were to graph these equations in an \( xy \)-coordinate system: 1. **Initial Observation**: - As \( t \) increases from 0 to \( \pi/2 \), \( x \) increases from 0 to 1 and \( y \) decreases from 1 to 0. - From \( \pi/2 \) to \( \pi \), \( x \) decreases from 1 to 0 and \( y \) increases from 0 to 1. - This behavior repeats in each interval of length \( \pi \). 2. **Graph Shape**: - The path described is a parabolic segment or a sinusoidal variation between two mirrored curves, owing to the equations involving squares of sine and cosine. - The entire path is symmetric across the lines \( x = y \) and \( y = x \). 3. **Repetitions**: - Given \( t \) runs from 0 to \( 3\pi \), this behavior repeats three times, cycling through from a start at the origin through peak points at \( (1, 0) \) and back to another symmetric point at \( (0, 1) \). Understanding these parametric equations through their graphical interpretation and algebraic manipulation can help in analyzing trigonometric behavior and the nature of such periodic functions.
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