Problem 20 - Calculus Early Transcendentals: Pearson New International Edition

Question

In each of Problems 1-20, a parametric representation of a curve is given. (a) Graph the curve. (b) Is the curve closed? Is it simple? (c) Obtain the Cartesian equation of the curve by eliminating the parameter (see Examples 1-4). $$ x=\sin \theta, y=2 \cos ^{2} 2 \theta ;-\infty<\theta<\infty $$

Step-by-Step Solution

Answer

The curve is closed and simple. The Cartesian equation simplifies as \( y = 2 \left(\frac{1 + \sqrt{1-x^2}}{2}\right)^2 \).

Step 1: Graphing the Curve

To graph the curve, recognize that \( x = \sin \theta \) represents a simple wave form. The expression for \( y \) is \( y = 2 \cos^2 2\theta \). By varying \( \theta \) from \(-\infty\) to \(\infty\), you can understand the curve by plotting selected points or using graphing technology. This curve will repeat every \( \pi \) due to both sine and cosine functions being periodic, which can help in visualizing the graph.

Step 2: Determine if the Curve is Closed and Simple

A closed curve means that it returns to its starting point and forms a loop. A simple curve is one that does not cross itself. The functions \( x = \sin \theta \) and \( y = 2 \cos^2 2\theta \) are periodic with periods \( 2\pi \) and \( \pi \) respectively. Thus, the curve is closed. To determine if it is simple, consider that within one period, the function does not retrace its path; hence, it is simple.

Step 3: Eliminate the Parameter

To find the Cartesian equation, eliminate the parameter \( \theta \). Start with \( x = \sin \theta \) and express \( \theta \) in terms of \( x \). Since \( \sin^2 \theta = 1 - \cos^2 \theta \), we find that \( x^2 = \sin^2 \theta \). The expression for \( y \) becomes \( y = 2 \cos^2 2\theta \) which can be rewritten using the double angle identity: \( \cos 2\theta = 2\cos^2 \theta - 1 \). Therefore, \( y = 2 \left(\frac{1 + \sqrt{1-x^2}}{2}\right)^2 \) is the resulting Cartesian equation for certain sections of the curve.

Key Concepts

Graphing Parametric Curves

Graphing parametric curves involves plotting points that correspond to certain values of a parameter, typically denoted as \( \theta \). For example, the curve given by \( x = \sin \theta \) and \( y = 2 \cos^2 2\theta \) represents a wave-like form for \( x \) and looks more complex for \( y \). When graphing this curve, you select values for \( \theta \), calculate the corresponding \( x \) and \( y \) values, and plot these points on a graph.

• Parametric equations produce points in the xy-plane by calculating positions as the parameter changes.
• Understanding the periodic nature of \( \sin \theta \) and \( \cos \theta \) is crucial for graphing, as they repeat every \( 2\pi \) and \( \pi \) respectively.
• Utilizing technology, such as graphing calculators or software, can aid in visualizing how the curve forms and repeats over intervals.

Using these principles will help you graph any curve defined by parametric equations.

Closed and Simple Curves

Determining if a curve is closed and simple involves examining its behavior over its period. A closed curve circles back to its start point to form a loop. A simple curve does not cross itself. In our example, the parametric equations are \( x = \sin \theta \) and \( y = 2 \cos^2 2\theta \), which indicate certain periodic behaviors.

• The periodic function \( x = \sin \theta \) completes its cycle every \( 2\pi \), and \( y = 2 \cos^2 2\theta \) completes its cycle every \( \pi \).
• This means that the curve repeats itself, confirming that it's closed.
• It is simple within one period because it doesn't intersect itself.

So, for the given parametric equations and their inherent periodicity, the curve is both closed and simple.

Cartesian Equation Derivation

Deriving the Cartesian equation from parametric equations involves eliminating the parameter, which in this case is \( \theta \). Start with \( x = \sin \theta \). To eliminate \( \theta \), use the identity \( \sin^2 \theta = 1 - \cos^2 \theta \), leading to \( x^2 = \sin^2 \theta \). For \( y = 2 \cos^2 2\theta \), apply the double angle identity, \( \cos 2\theta = 2\cos^2 \theta - 1 \), which helps in expressing \( y \) in terms of \( x \):- Substitute \( \cos \theta \) in terms of \( x \) using \( \cos^2 \theta = 1 - x^2 \).Thus, \( y = 2 \left(\frac{1 + \sqrt{1-x^2}}{2}\right)^2 \) represents the Cartesian equation for parts of the curve.By this method, parametric equations can be converted to Cartesian equations, allowing for interpretation and analysis in the familiar xy-plane.

Trigonometric Identities in Parametrization

Trigonometric identities are essential in simplifying parametric equations to derive Cartesian forms or to understand the behavior of curves. In our example, the identities \( \sin^2 \theta + \cos^2 \theta = 1 \) and \( \cos 2\theta = 2 \cos^2 \theta - 1 \) are key.

• These identities help in expressing parametric equations without the parameter, easing the derivation of Cartesian equations.
• Understanding how identities like the double angle formulas change the functions is crucial in predicting the shapes and nature of curves.
• They also aid in solving for \( y \) in terms of \( x \), bypassing direct dependency on the parameter \( \theta \).

Grasping these identities fundamentally connects trigonometry to coordinate geometry, improving how you approach problems involving parametric curves.