Problem 20 - Precalculus with Limits

Question

In Exercises 7-26, (a) sketch the curve represented by the parametric equations (indicate the orientation of the curve) and (b) eliminate the parameter and write the corresponding rectangular equation whose graph represents the curve. Adjust the domain of the resulting rectangular equation if necessary. \(x= \cos\ \theta\) \(y=2\ \sin\ 2\theta\)

Step-by-Step Solution

Answer

The final rectangular equation is \( 16x^4 + y^2 - 16x^2 = 0 \) for \( -1 \le x \le 1 \).

Step 1: Plotting the curve

Start by plotting some points to understand the curve better. For this, take some values of the parameter \( \theta \) and calculate the corresponding x and y values. Subsequently, plot these points on a graph. The direction in which increasing values of \( \theta \) move is the orientation of the curve.

Step 2: Eliminating the parameter

Use the double angle identity \( \sin 2\theta = 2 \sin \theta \cos \theta \). So, \( y = 2 \sin 2\theta \) can be written as \( y = 4 \sin \theta \cos \theta \). Now, since \( x = \cos \theta \), our equation becomes \( y = 4x \sin \theta \). Squaring both sides gives \( y^2 = 16x^2 \sin^2 \theta \). Since \( \sin^2 \theta = 1 - \cos^2 \theta = 1 - x^2 \), we put this value into the equation and get \( y^2 = 16x^2 (1 - x^2) = 16x^2 - 16x^4 \).

Step 3: Finalizing the equation

After rearranging, the rectangular equation turns out to be \( 16x^4 + y^2 - 16x^2 = 0 \). The domain should span all valid x-values for the original parametric equations. In this case, since \( x = \cos \theta \), x is between -1 and 1, inclusive. So, our final rectangular equation with adjusted domain is \( 16x^4 + y^2 - 16x^2 = 0 \), for \( -1 \le x \le 1 \).

Key Concepts

Sketching Curves

Sketching curves defined by parametric equations is a fundamental skill in mathematics that involves drawing the path of a point whose position is determined by parameters, typically along the x and y-axes. To sketch a curve, one must calculate the coordinates of several points by substituting different values of the parameter into the given equations. Plot these points on a graph and connect them smoothly to reveal the shape of the curve.

For instance, with parametric equations such as \( x = \cos \theta \) and \( y = 2 \sin 2\theta \), you begin by calculating the x and y values for various angles \(\theta\). It's also crucial to note the orientation, which you can determine by observing the direction in which the curve progresses as \(\theta\) increases. This process provides a visual understanding of the relationship between x and y, beyond what the equations alone can convey.

Eliminating Parameters

To convert parametric equations into a single rectangular equation, a process known as eliminating the parameter is employed. This turns a pair of equations involving a third variable into one equation in terms of x and y only. To achieve this, we use algebraic manipulations and identities to express all variables in terms of x and y.

For example, with the parametric equations \( x = \cos \theta \) and \( y = 2 \sin 2\theta \), you can use trigonometric identities—such as the double angle formula \( \sin 2\theta = 2 \sin \theta \cos \theta \)—to write y entirely in terms of x and a trigonometric function of \(\theta\). Then, by squaring and further algebraic manipulation, we can eliminate \(\theta\) to produce a rectangular equation. This simplification allows us to represent the curve with an equation that can be graphed on the Cartesian plane without parameter \(\theta\).

Rectangular Equation

A rectangular equation is an expression that relates x and y coordinates on a Cartesian plane without involving a third parameter. It's the conventional form of an equation for graphing purposes. The transition from parametric equations to a rectangular form is desirable for various applications, including solving systems of equations or integrating to find area.

In our case, by squaring and applying trigonometric identities, we transformed the parametric equations into the rectangular form \( 16x^4 + y^2 - 16x^2 = 0 \). This equation should be adjusted for the domain, which is the set of all possible x-values. The domain of the original parametric equation is constrained by the range of \( \cos \theta \), which is \(-1 \leq x \leq 1\). The rectangular equation, with its domain duly noted, completely describes the curve in a plane without reference to the parametric variable \(\theta\).

Double Angle Identities

Double angle identities are trigonometric relationships that involve the sine, cosine, or tangent of twice a particular angle. They are especially useful in simplifying expressions where the angle is doubled or when eliminating the parameter in parametric equations. The most common double angle identities are \( \sin 2\theta = 2\sin \theta \cos \theta \) and \( \cos 2\theta = \cos^2 \theta - \sin^2 \theta \), but they can also be expressed in terms of \( \cos^2 \theta \) or \( \sin^2 \theta \) alone.

In our exercise, to eliminate the parameter and obtain a rectangular equation, we used the identity for \(\sin 2\theta\) to express y in terms of x and \(\sin\theta\), which was then squared to involve \(\sin^2 \theta\). The double angle identity allowed us to find an equation that relates x and y directly, thus enabling us to sketch the curve accurately without the original parameter \(\theta\).