Problem 26 - Precalculus

Question

(A) sketch the curve represented by the parametric equations (indicate the orientation of the curve) and (b) eliminate the parameter and write the resulting rectangular equation whose graph represents the curve. Adjust the domain of the rectangular equation, if necessary. $$\begin{aligned} &x=\cos \theta\\\ &y=2 \sin 2 \theta \end{aligned}$$

Step-by-Step Solution

Answer

The graph of the parametric equations, \( x = \cos \theta \) and \( y = 2 \sin 2 \theta \), forms a curve which is typically visualized via a graphing tool. After eliminating the parameter \( \theta \), a rectangular equation is obtained: \( y = 2x\sqrt{1 - x^2} \) which is valid over \( -1 \le x \le 1 \).

Step 1: Visualization of the curve

The equations provided are parametric equations for \( x \) and \( y \) with parameter \( \theta \). The first equation given is \( x = \cos \theta \). The second equation provided is \( y = 2 \sin 2 \theta \). The parametric equations then trace out a curve as \( \theta \) ranges over an interval. To visualize it, think of \( \theta \) as time and \( (x, y) \) as the position of a particle at time \( \theta \). Use a graphing tool or calculator to sketch the graph of these equations.

Step 2: Eliminating the parameter

To eliminate the parameter \( \theta \) call upon the trigonometric identities: \( 2 \sin \theta \cos \theta = \sin 2 \theta \). Then from the first equation, we solve for \( \cos \theta \), which is \( x \). We can substitute this into the expression \( 2 \sin 2 \theta \) in the second equation. Thus \( y = 2 \sin 2 \theta = 2 \sin \theta (2\cos \theta) \). Since \( \sin \theta = \sqrt{1 - \cos^2 \theta} \) (using the Pythagorean identity for sine and cosine), we substitute \( x \) in place of \( cos \theta \) again, ending up with a single rectangular equation: \( y = 2x\sqrt{1 - x^2} \).

Step 3: Adjust the domain of the rectangular equation

To adjust the domain, take into account that the equation \( y = 2x\sqrt{1 - x^2} \) is valid for \( -1 \le x \le 1 \) because of the square root function and the range of the cosine function.

Key Concepts

Trigonometric Identities

Trigonometric identities are like helpful shortcuts that relate different trigonometric functions to each other. They can make solving equations simpler by letting us replace expressions with something we already know, like a more familiar angle or function. In this exercise, we use the trigonometric identity \( 2 \sin \theta \cos \theta = \sin 2\theta \). This is called the double-angle identity for sine, and it allows us to connect the parameter \( \theta \) with our rectangular equation.

• Cosine Identity: It expresses \( \cos 2\theta \) in terms of \( \cos \theta \) and \( \sin \theta \) using \( \cos^2 \theta + \sin^2 \theta = 1 \).
• Solution Approach: Allows expressing one variable through another, such as using \( \cos \theta = x \) to substitute into other identities.

Understanding how these identities work will enable you to eliminate the parameter effectively, moving from parametric to rectangular forms, which shows the function in terms of \( x \) and \( y \). This ultimately helps you sketch the curve and reveal its path.

Rectangular Equations

Rectangular equations are your typical equations that relate \( x \) and \( y \) in the Cartesian plane. Unlike parametric equations, they don't involve a third parameter, and usually, they show the relationship directly between the two variables.When we start with parametric equations like \( x = \cos \theta \) and \( y = 2 \sin 2\theta \), we can eliminate \( \theta \) to find a rectangular equation.

Steps to Derive Rectangular Equations:

- First, express one parameter in terms of the other using identities or operations that simplify the equation.- Use the identity \( \sin \theta = \sqrt{1 - \cos^2 \theta} \).- Substitute and simplify the steps to bring one equation into terms of the other, like substituting \( \cos \theta = x \) into our equation for \( y \).Ultimately, we obtained the equation \( y = 2x\sqrt{1 - x^2} \), which perfectly describes the path traced by the parametric equations without directly involving \( \theta \). This is especially useful for graphing the curve in a straightforward manner.

Domain of Functions

The domain of a function specifies all the possible values of \( x \) for which the function is defined. When converting a parametric equation into a rectangular form, understanding the domain is crucial.During this exercise, when we derived the rectangular equation \( y = 2x\sqrt{1 - x^2} \), we needed to carefully adjust the domain. This equation involves a square root, and the expression under the root must be non-negative, meaning \( 1 - x^2 \geq 0 \). Therefore, the values of \( x \) range from \(-1\) to \(1\) since \( \cos \theta \) varies between these values by definition and affect the domain naturally.

• Ensure the square root expression is non-negative: \( 1 - x^2 \geq 0 \) leads to \( -1 \le x \le 1 \).
• The natural range of trigonometric functions like cosine also influences the adjusted domain.

Recognizing the domain helps prevent misleading graphs or undefined calculations, and it fine-tunes your interpretation of function behavior over specific intervals.