Problem 25 - 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. $$\begin{aligned} &x=6 \sin 2 \theta\\\ &y=6 \cos 2 \theta \end{aligned}$$

Step-by-Step Solution

Answer

The given parametric equations represent a circle of radius 6 centered at the origin, traversed twice in an anti-clockwise direction for every cycle of \(2\pi\) in \(\theta\). When the parameter \(\theta\) is eliminated, the rectangular equation of the curve is \[x^2 + y^2 = 36\].

Step 1: Sketching the Curve

Using the idea that \[x=r \cos(\theta)\] and \[y=r \sin(\theta)\] represent a circle with radius \(r\) in a polar coordinate, we can view the given equations \(x = 6\sin2\theta\) and \(y = 6\cos2\theta\) as representing a circle of radius 6 in the Cartesian coordinate system. The '2' in both sine and cosine functions indicates that the circle will be traversed twice for every cycle of \(2\pi\) in \(\theta\), giving the orientation in the anti-clockwise direction.

Step 2: Eliminating the Parameter

To eliminate the parameter \(\theta\), square both equations and add them, using the Pythagorean trigonometric identity \[\sin^2 \theta + \cos^2 \theta = 1\]. Squaring \[x = 6\sin2\theta\] gives \[x^2 = 36\sin^22\theta\] and squaring \[y = 6\cos2\theta\] gives \[y^2 = 36\cos^22\theta\]. Adding these two equations gives \[x^2 + y^2 = 36(\sin^22\theta + \cos^22\theta) = 36 \times 1 = 36\]. This is the rectangular form of the equation for the curve, which describes a circle of radius 6 centered at the origin.

Key Concepts

Rectangular Equations

Rectangular equations are one of the most common ways to describe curves on the Cartesian plane. In this context, they relate the variables \(x\) and \(y\) through an equation, eliminating any parameters, such as \(\theta\). In our exercise, the goal is to convert from parametric equations to a rectangular equation.
When working with parametric equations, like \(x = 6\sin2\theta\) and \(y = 6\cos2\theta\), the process involves eliminating the parameter to express \(x\) and \(y\) directly in terms of each other. This is typically done by using algebraic manipulations along with trigonometric identities. For the given equations, we square each and sum them up, aided by the identity \(\sin^2\theta + \cos^2\theta = 1\).
By squaring and adding to remove \(\theta\), we arrive at the equation \(x^2 + y^2 = 36\). This rectangular equation corresponds to a circle with a radius of 6 centered at the origin. It's a classic approach to translating between different forms of representing curves.

Polar Coordinates

Polar coordinates offer a unique way to describe points and curves, particularly useful for circular and spiral paths. Instead of using \(x\) and \(y\), they use the radius \(r\) (how far you are from the origin) and the angle \(\theta\) (your direction from the positive x-axis).
In our task, both the parametric equations \(x = 6\sin2\theta\) and \(y = 6\cos2\theta\) are initially expressed in a fashion that aligns well with polar coordinates. Given their forms, one can envision \(r = 6\) as these represent parameterized versions of a circle's components, modified by function transformations that introduce oscillations due to \(2\theta\).
This polar representation provides insights into how the circle forms and at what speed it is traversed. The doubling of \(\theta\), shown in the parametric equations, means that each full cycle of \(2\pi\) in \(\theta\) results in completing the circle twice. Understanding polar coordinates helps to visualize complex parametric paths and their orientation.

Trigonometric Identity

Trigonometric identities are essential tools in both algebra and calculus. They help transform and simplify expressions involving trigonometric functions. A fundamental identity used in our context is \(\sin^2\theta + \cos^2\theta = 1\). This identity aids in converting parametric forms to rectangular forms, as seen in our exercise.
When converting \(x = 6\sin2\theta\) and \(y = 6\cos2\theta\) to a rectangular equation, we utilize this identity after squaring both parametric equations: \(x^2 = 36\sin^22\theta\) and \(y^2 = 36\cos^22\theta\). Adding these up, we recognize that \(\sin^22\theta + \cos^22\theta = 1\) simplifies the equation to \(x^2 + y^2 = 36\).
This simplifies to a neat rectangular equation, \(x^2 + y^2 = 36\), illustrating a circle with a radius of 6. Trigonometric identities like this help streamline the conversion process and reveal deeper relationships in complex mathematical tasks.