Problem 20 - Calculus Early Transcendentals: Pearson New International Edition

Question

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

Step-by-Step Solution

Answer

The parametric curve ranges through \((-1, 1)\) for \(x\) and \((0, 2)\) for \(y\).

Step 1: Understanding Parametric Equations

The given curve is represented by the parametric equations \( x = \sin \theta \) and \( y = 2 \cos^2 (2\theta) \). Here, \( \theta \) is the parameter which varies from \(-\infty\) to \(\infty\). Our goal is to understand how \( x \) and \( y \) change as \( \theta \) changes.

Step 2: Expressing Trigonometric Terms

First, identify \( x \) in terms of \( \theta \). We have \( x = \sin \theta \). For \( y \), express \( \cos^2(2\theta) \) using the identity \( \cos^2 \alpha = \frac{1 + \cos(2\alpha)}{2} \), so \( y = 2\cos^2(2\theta) = 1 + \cos(4\theta) \).

Step 3: Connecting the Equations

Now, use the identity \( \sin^2\theta + \cos^2\theta = 1 \) to express \( \cos^2\theta \) in terms of \( x \). From \( x = \sin \theta \), we have \( \cos^2\theta = 1 - x^2 \). But \( y = 1 + \cos(4\theta) \) doesn't directly link to \( x \).

Step 4: Verifying and Interpreting the Equations

Since an explicit relation wasn't deduced from \( y = 1 + \cos(4\theta) \) in simple terms of \( x \) without further trigonometric transformations, let’s derive equivalence: \( \cos(4\theta) \) could range from \(-1\) to \(1\), thus \( y \) ranges from \(0\) to \(2\). Therefore, \( y \) varies without direct dependence on \( x \).

Step 5: Analyzing the Curve Behavior

The behavior of the curve can be analyzed based on \( x = \sin \theta \) which ranges from \(-1\) to \(1\), and \( y = 1 + \cos(4\theta) \) ranges from \(0\) to \(2\). The curve periodically traces over these ranges as \( \theta \) varies through its domain.

Key Concepts

Trigonometric Identities

Trigonometric identities are keys to unlocking transformations and simplifications of equations involving trigonometric functions. They provide a set of equations that hold true for all values and help simplify expressions and find connections between functions.
For instance, the identity \( \cos^2 \alpha = \frac{1 + \cos(2\alpha)}{2} \) is particularly useful in dealing with squared cosine terms. In our problem, this identity allows the transformation of \( \cos^2(2\theta) \) into a form involving \( \cos(4\theta) \).

• Using this identity, \( y = 2\cos^2(2\theta) = 1 + \cos(4\theta) \) is derived, transforming a harder-to-manage term into a simpler expression.
• These transformations help in matching equations and discovering the behavior of the curve by revealing underlying symmetries or periodicities.

As you learn and practice using these identities, they become invaluable tools in your mathematical toolkit, making complex trigonometry much more manageable.

Curve Behavior

Understanding curve behavior involves investigating how a curve responds to changes in its parameters. In parametric equations, this means looking at how both \( x \) and \( y \) vary as the parameter, in this case \( \theta \), changes.
The provided equations, \( x = \sin \theta \) and \( y = 1 + \cos(4\theta) \), show us different ranges for these variables:

• \( x = \sin \theta \) ranges from \(-1\) to \(1\), as the sine function oscillates between these values due to its periodic nature.
• \( y = 1 + \cos(4\theta) \) has values from \(0\) to \(2\), derived from \(\cos(4\theta)\) oscillating from \(-1\) to \(1\).

As \( \theta \) moves from \(-\infty \) to \(\infty \), these trigonometric functions describe a repeating cycle due to their inherent periodic properties.
This cyclical nature implies that the visual representation of the curve returns to similar shapes repeatedly, characterizing its behavior over its domain. Familiarity with trigonometric function behavior aids in predicting how changes to \( \theta \) will influence the entire curve.

Trigonometric Transformations

Trigonometric transformations entail manipulating trigonometric expressions to simplify, solve, or better understand them. These transformations are integral in converting more complex forms into manageable ones.
In dealing with parametric equations, such as our exercise, a significant transformation was expressing \( y = 2\cos^2(2\theta) \) into \( y = 1 + \cos(4\theta) \) using a trigonometric identity.

• This transformation simplifies the analysis, providing a straightforward range for \( y \) while maintaining the essence of the initial expression.
• It also helps identify and verify the periodic nature of \( y \), which is crucial for understanding how it behaves as \( \theta \) changes.

Through such transformations, the relationships between angles and functions are better clarified, revealing insights into behaviors and properties of trigonometric equations in various contexts.
Mastering these transformations enhances your ability to navigate through complex trigonometric scenarios, crucial in both academic and real-world applications.