Problem 19 - Calculus Early Transcendentals: Pearson New International Edition

Question

a parametric representation of a curve is given. $$ x=\cos \theta, y=-2 \sin ^{2} 2 \theta ;-\infty<\theta<\infty $$

Step-by-Step Solution

Answer

The Cartesian equation is \( y = -8x^2 + 8x^4 \).

Step 1: Understand Parametric Equations

The given parametric equations represent a curve. The functions for the curve are given by \( x = \cos \theta \) and \( y = -2 \sin^2 2\theta \). Our goal is to eliminate the parameter \( \theta \) to find the Cartesian equation of the curve.

Step 2: Use Trigonometric Identity

We know from trigonometric identities that \( \cos^2\theta + \sin^2\theta = 1 \). This implies that \( \sin^2\theta = 1 - \cos^2\theta \).

Step 3: Express \( \sin^2 2\theta \)

Using the double angle identity, \( \sin 2\theta = 2 \sin \theta \cos \theta \), we get \( \sin^2 2\theta = 4\sin^2\theta\cos^2\theta \).

Step 4: Substitute \( x = \cos \theta \) and \( \sin^2\theta = 1-x^2 \)

Express \( \sin^2 2\theta \) in terms of \( x \) using \( \sin 2\theta = 2 \sin \theta \cos \theta = 2 \sqrt{1-x^2}x \). Thus, \( \sin^2 2\theta = 4x^2(1-x^2) \).

Step 5: Substitute \( \sin^2 2\theta \) into \( y \)

Using \( y = -2 \sin^2 2\theta \), substitute \( \sin^2 2\theta = 4x^2(1-x^2) \), resulting in \( y = -2[4x^2(1-x^2)] = -8x^2 + 8x^4 \).

Step 6: Write the Cartesian Equation

The Cartesian equation derived from the given parametric equations is \( y = -8x^2 + 8x^4 \).

Key Concepts

Cartesian Equation

When dealing with parametric equations, our main task often involves finding the Cartesian equation. A Cartesian equation encourages us to express a relationship between two variables, usually eliminating any parameters involved.
To convert from parametric to Cartesian, we start by understanding each individual equation. One equation represents one variable, such as

• using a cosine or sine function for one of the parameters.


• Then, by using connections between these parameters, we eliminate them to simplify the relationship into a conventional form like \( y = f(x) \).

For this exercise, we see, by evaluating the parametric equations, that the connection between the equations involves trigonometric identities. The ultimate goal is to find the expression of \( y \) in terms of \( x \) only, resulting in the Cartesian equation \( y = -8x^2 + 8x^4 \). Once these steps are followed, the resultant equation captures the essence of the curve in a single algebraic equation.

Trigonometric Identities

Trigonometric identities are fundamental tools in transforming and simplifying equations involving trigonometric functions. They help us understand and manipulate angles and their relationships.
Key identities:

• \( \sin^2\theta + \cos^2\theta = 1 \): This identity allows us to substitute \( \sin^2\theta = 1 - \cos^2\theta \) and vice versa.


• Various angle addition and subtraction formulas.

In this exercise, trigonometric identities enable the expression \( \sin^2\theta \) in terms of \( \cos\theta \). By using identity transformations, we explore the relationship between different angles and transform their algebraic form. This is crucial in parametric equations where we aim to form a relation independent of parameters like \( \theta \). Through these transformations, solving complex trigonometric equations becomes significantly more manageable.

Double Angle Identity

The double angle identity is pivotal in simplifying expressions where angles are effectively doubled: \( 2\theta \), \( 3\theta \), etc. Specifically, the double angle identity for sine is:

• \( \sin 2\theta = 2\sin\theta\cos\theta \)

This identity assists in solving equations by breaking down complex expressions into more understandable components. When squaring this form, as seen here, \( \sin^2 2\theta \) can be represented as \( 4\sin^2\theta\cos^2\theta \). In our exercise:

• By inserting the expressions resulting from the double angle identity into the parametric equations, simplification is achieved for \( y \) in terms of \( x \).


• We reexpress our equation to get \( y = -2[4x^2(1-x^2)] = -8x^2 + 8x^4 \).

Using these identities reduces complex trigonometric expressions into accessible forms, letting us more easily derive the Cartesian equation, reflecting a clear relationship between \( x \) and \( y \).