Problem 15 - Algebra and Trigonometry

Question

\begin{array}{l}{1-22 \text { a pair of parametric equations is given. }} \\\ {\text { (a) Sketch the curve represented by the parametric equations. }} \\\ {\text { (b) Find a rectangular-coordinate equation for the curve by }} \\\ {\text { eliminating the parameter. }}\end{array} $$ x=\cos t, \quad y=\cos 2 t $$

Step-by-Step Solution

Answer

Sketch: Lissajous figure. Rectangular equation: \( y = 2x^2 - 1 \).

Step 1: Understand the Parametric Equations

The given parametric equations are \( x = \cos t \) and \( y = \cos 2t \). The parameter \( t \) could represent an angle, hence as \( t \) changes, both \( x \) and \( y \) change, tracing out a curve in the xy-plane.

Step 2: Sketch the Curve by Finding Points

Evaluate the equations for several values of \( t \). For example, when \( t = 0 \), \( x = 1 \) and \( y = 1 \); when \( t = \frac{\pi}{4} \), \( x = \frac{\sqrt{2}}{2} \) and \( y = 0 \); when \( t = \frac{\pi}{2} \), \( x = 0 \) and \( y = -1 \). Plot these points to get an idea of the curve's shape. Continue this process to see the shape more clearly. Typically, this curve will trace out a path known as a Lissajous figure.

Step 3: Eliminate Parameter to Find Rectangular Equation

Using the identity \( \cos 2t = 2 \cos^2 t - 1 \), express \( \cos 2t \) in terms of \( \cos t \). Substitute \( x = \cos t \) into this identity: \( y = 2x^2 - 1 \). This equation represents the curve in rectangular form.

Step 4: Write the Rectangular Equation

The rectangular form of the equations is \( y = 2x^2 - 1 \). This is a parabola that opens downwards.

Key Concepts

Rectangular Coordinate System

The rectangular coordinate system, often referred to as the Cartesian plane or xy-plane, is foundational in algebra and geometry. It's a two-dimensional plane defined by two perpendicular axes: the x-axis and the y-axis. These axes divide the plane into four quadrants where points are located based on ordered pairs \((x, y)\).

In the given problem, parametric equations \(x = \cos t\) and \(y = \cos 2t\) tell us how to locate points on this plane. As the parameter \(t\) varies, points are plotted on the plane, and their collection traces the curve. Therefore, every point on this curve is an intersection of specific \(x\) and \(y\) values.

• The x-axis typically represents a horizontal line where values increase from left to right.
• The y-axis denotes a vertical line where values increase from bottom to top.

Understanding how to plot points in this system is crucial for sketching curves represented by parametric equations.

Lissajous Figures

Lissajous Figures are complex geometric curves, often created when plotting parametric equations that involve trigonometric functions. These figures are characterized by their intricate loops formed by two perpendicular harmonic oscillations, which explain their aesthetic appeal.

In the exercise, the equations \(x = \cos t\) and \(y = \cos 2t\) produce a Lissajous Figure. When you plot these on the rectangular coordinate system, you notice interesting patterns and loops. They visually represent the relationship between two oscillations of different frequencies. The curve's shape and complexity depend on the ratio between these two frequencies.

• If the frequency ratio is simple, such as 1:2 (as in our problem), the figure is more predictable.
• Lissajous Figures can also help visualize sound waves and various engineering applications due to their oscillatory nature.

Exploring Lissajous Figures widens the understanding of both mathematics and physics since it combines harmonic motion with visual geometry.

Trigonometric Identities

Trigonometric identities are equations that relate various trigonometric functions together. They play a critical role when working with parametric equations and converting them into rectangular form. These identities simplify calculations and allow us to derive useful mathematical insights.

In the solution, the trigonometric identity \(\cos 2t = 2 \cos^2 t - 1\) is used to express \(y\) in terms of \(x\). By substituting \(x = \cos t\) into this identity, the rectangular form of the equation \(y = 2x^2 - 1\) is achieved. This conversion is essential in understanding the curve's behavior in a straightforward visual context.

• Trigonometric identities often involve sine, cosine, and tangent functions.
• They are essential for transforming equations to make them easier to work with or to provide clearer insights into the problem at hand.

Grasping these identities enhances problem-solving skills across various mathematical topics, from calculus to geometry.