Problem 13 - Calculus Early Transcendentals: Pearson New International Edition

Question

a parametric representation of a curve is given. $$ x=2 \sin t, y=3 \cos t ; 0 \leq t \leq 2 \pi $$

Step-by-Step Solution

Answer

The parametric curve represents an ellipse \( \frac{x^2}{4} + \frac{y^2}{9} = 1 \).

Step 1: Recognize the Parametric Equations

We are given two parametric equations, \( x = 2 \sin t \) and \( y = 3 \cos t \), describing a curve in terms of the parameter \( t \). The parameter \( t \) is varying from \( 0 \) to \( 2\pi \). Our goal is to express this relationship in terms of \( x \) and \( y \).

Step 2: Eliminate the Parameter

To eliminate the parameter \( t \), we use trigonometric identities. We know from trigonometry that \( \sin^2 t + \cos^2 t = 1 \). We will apply this identity to the parametric equations given. Start by isolating \( \sin t \) and \( \cos t \) in the equations:\( \sin t = \frac{x}{2} \) and \( \cos t = \frac{y}{3} \).

Step 3: Apply Trigonometric Identity

Substitute the expressions for \( \sin t \) and \( \cos t \) into the identity \( \sin^2 t + \cos^2 t = 1 \):\[ \left(\frac{x}{2}\right)^2 + \left(\frac{y}{3}\right)^2 = 1 \]

Step 4: Simplify the Equation

Simplify \( \left(\frac{x}{2}\right)^2 + \left(\frac{y}{3}\right)^2 = 1 \) to find:\[ \frac{x^2}{4} + \frac{y^2}{9} = 1 \] This equation is the standard form of an ellipse.

Key Concepts

Ellipse

An ellipse is a geometric shape that looks like an elongated circle. It is defined as the set of all points for which the sum of the distances to two fixed points, called foci, is a constant. In the context of Cartesian coordinates, an ellipse can be described by the standard equation:\[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \]where \(a\) and \(b\) are the lengths of the semi-major and semi-minor axes, respectively. The semi-major axis is always the longest, stretching horizontally or vertically, depending on the orientation of the ellipse. In our exercise, the semi-major axis is vertical, defined by the value 3 from the \(y\) equation, and the semi-minor axis is horizontal, given by the value 2 from the \(x\) equation.Ellipses have interesting properties such as the reflective property, where paths emanating from one focus reflect and pass through the other focus. They are commonly found in the orbits of planets and moons as well as various engineering designs. Recognizing the standard form of an ellipse equation is crucial for understanding how it relates to its parametric equations.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for every value of the occurring variables, where they are defined. They play a significant role in simplifying expressions and solving equations, especially when dealing with parametric forms.One key identity used in this exercise is:- \( \sin^2 t + \cos^2 t = 1 \)This fundamental identity stems from the Pythagorean theorem applied in the unit circle, where \(t\) represents an angle. It shows the intrinsic relationship between the sine and cosine of an angle, describing a circle in the coordinate plane.In our scenario, this identity allows us to connect \(\sin\) and \(\cos\) terms derived from the given parametric equations: \(x = 2 \sin t\) and \(y = 3 \cos t\). By manipulating these, we can transition from parametric equations to a rectangular form, facilitating better understanding and solving of the problem.

Parameter Elimination

Parameter elimination involves removing the parameter \(t\) from parametric equations to express the relationship in terms of \(x\) and \(y\) directly. It is a vital step in converting parametric equations into a more familiar Cartesian format, which is often easier to interpret and analyze.The process begins by isolating \(\sin t\) and \(\cos t\) in the equations:- \(\sin t = \frac{x}{2}\)- \(\cos t = \frac{y}{3}\)Using the trigonometric identity \( \sin^2 t + \cos^2 t = 1 \), substitute the isolated expressions into the identity:\[ \left(\frac{x}{2}\right)^2 + \left(\frac{y}{3}\right)^2 = 1 \]Through algebraic manipulation, this simplifies to \( \frac{x^2}{4} + \frac{y^2}{9} = 1 \), yielding the equation of an ellipse. Understanding parameter elimination is crucial for solving problems where parametric relationships need to be expressed in a different form to interpret or solve them more effectively.