Problem 17 - Calculus Early Transcendentals: Pearson New International Edition

Question

a parametric representation of a curve is given. $$ x=9 \sin ^{2} \theta, y=9 \cos ^{2} \theta ; 0 \leq \theta \leq \pi $$

Step-by-Step Solution

Answer

The Cartesian equation is \( x + y = 9 \).

Step 1: Identify Parametric Equations

The given parametric equations are: \( x = 9 \sin^2 \theta \) and \( y = 9 \cos^2 \theta \). These describe the coordinates \( (x, y) \) of a point on the curve as functions of the parameter \( \theta \), where \( 0 \leq \theta \leq \pi \).

Step 2: Express Trigonometric Identity

Recall the trigonometric identity \( \sin^2 \theta + \cos^2 \theta = 1 \). This is crucial in relating \( x \) and \( y \) without the parameter \( \theta \).

Step 3: Substitute and Simplify

Express \( \sin^2 \theta \) and \( \cos^2 \theta \) in terms of \( x \) and \( y \): \( \sin^2 \theta = \frac{x}{9} \) and \( \cos^2 \theta = \frac{y}{9} \). Substitute these into the identity: \( \frac{x}{9} + \frac{y}{9} = 1 \).

Step 4: Solve for Cartesian Equation

Multiply the entire equation by 9 to eliminate the fractions: \( x + y = 9 \). This is the Cartesian equation representing the given parametric curve.

Key Concepts

Trigonometric Identities

Understanding trigonometric identities is essential when working with parametric equations. These identities provide a way to relate different trigonometric functions, such as \[ \sin \theta \] and \[ \cos \theta \], to one another. One of the most fundamental identities is \[ \sin^2 \theta + \cos^2 \theta = 1 \]. This relationship is derived from the circle in trigonometry and is valid for any angle \( \theta \).
In the context of parametric equations, trigonometric identities allow us to eliminate the parameter \( \theta \) when necessary. This makes it easier to transition from a parametric form to a more universally used Cartesian form. By squaring the sine and cosine components in the given parametric equations, and recognizing their sum equals 1, we prepare the equations for transformation.
Identifying the right trigonometric identity to use is crucial for solving problems and can simplify many mathematical expressions.

Cartesian Equation

When dealing with parametric equations, it's often necessary to reformulate them into a Cartesian equation. A Cartesian equation describes a curve in the \( xy \)-plane without using a parameter. It typically results from eliminating the parameter from the given parametric forms.
For the given problem, the parametric equations \( x = 9 \sin^2 \theta \) and \( y = 9 \cos^2 \theta \) need to be combined into one equation. Using the trigonometric identity \( \sin^2 \theta + \cos^2 \theta = 1 \), we substitute \( \sin^2 \theta = \frac{x}{9} \) and \( \cos^2 \theta = \frac{y}{9} \) into the identity. This leads to the equation \( \frac{x}{9} + \frac{y}{9} = 1 \).
Next, by multiplying the entire equation by 9, we simplify it to \( x + y = 9 \). This linear equation is the Cartesian form of the parametric equations. The process not only simplifies the expression but provides a clear geometric representation of the curve.

Curve Representation

Representing curves via parametric equations involves defining both \( x \) and \( y \) as functions of a third variable, often known as the parameter \( \theta \). This third variable typically ranges over a specified interval, as in \( 0 \leq \theta \leq \pi \) in the problem at hand.
Parametric representations can offer more flexibility and expressiveness compared to Cartesian equations because they allow for the efficient tracing of curves which may not be easily expressible as a single function in \( x \) or \( y \). In the given exercise, the parameter \( \theta \) tracks the movement along a segment of a particular curve.
By transitioning from the parametric form to the Cartesian equation \( x + y = 9 \), we visualize the curve as a line segment in the Cartesian plane. Understanding and switching between these two forms of curve representation is an invaluable skill, as it enables deeper insights into the geometrical and functional behavior of curves.