Problem 15 - Precalculus: Mathematical for Calculus

Question

A pair of parametric equations is given. (a) Sketch the curve represented by the parametric equations. (b) Find a rectangular-coordinate equation for the curve by eliminating the parameter. $$x=\cos t, \quad y=\cos 2 t$$

Step-by-Step Solution

Answer

(a) Sketch a plot in [-1,1] for both x and y. (b) The equation is \( y = 2x^2 - 1 \).

Step 1: Understanding Parametric Equations

The given parametric equations are \( x = \cos t \) and \( y = \cos 2t \). The parameter \( t \) can be any real number, representing an angle in radians.

Step 2: Sketching the Parametric Curve

To sketch the curve, note that both \( x = \cos t \) and \( y = \cos 2t \) are periodic functions. \( x \) ranges from -1 to 1 as \( t \) varies and has a period of \( 2\pi \). \( y = \cos 2t \) has a period of \( \pi \) and also ranges from -1 to 1. By plotting points or recognizing trigonometric identities, sketch a pattern within these bounds.

Step 3: Expressing \( \cos 2t \) as a Function of \( x \)

Using the double angle identity for cosine, \( \cos 2t = 2\cos^2 t - 1 \). Since \( x = \cos t \), substitute \( x \) into this identity: \( y = 2x^2 - 1 \).

Step 4: Writing the Rectangular Equation

The rectangular-coordinate equation of the curve is derived by expressing \( y \) solely in terms of \( x \). From Step 3, we found that \( y = 2x^2 - 1 \). Therefore, the rectangular equation of the curve is \( y = 2x^2 - 1 \).

Key Concepts

Rectangular Coordinates

The rectangular coordinate system, also known as the Cartesian coordinate system, is foundational in plotting equations and functions. In this system, every point in a plane is identified by two numbers, usually called the x-coordinate and the y-coordinate. These represent the horizontal and vertical distances of the point from the origin, respectively.

In the context of parametric equations, a rectangular-coordinate equation represents the combination of parametric equations into a single equation, eliminating the parameter. For example, if you have parametric equations like \( x = \cos t \) and \( y = \cos 2t \), you can attempt to eliminate t to express y directly in terms of x. This makes plotting and understanding the overall shape of the curve easier without considering the parameter.

This transformation helps to simplify checking intersections, defining domain and range, and applying calculus techniques in rectangular form.

Trigonometric Identities

Trigonometric identities are crucial tools in mathematics, particularly when working with parametric equations. These identities are equations that hold for all values of the involved variables where both sides of the equation are defined.

For example, common trigonometric identities include the Pythagorean identity \( \sin^2\theta + \cos^2\theta = 1 \) and the double angle formula \( \cos 2\theta = 2\cos^2\theta - 1 \). In the problem given, the identity used is the latter. This helps in expressing one trigonometric function in terms of another, which is useful for eliminating parameters.

By rearranging and substituting into parametric equations, identities allow you to convert parametric equations into a more standard algebraic form. Thus, they are indispensable in both simplifying expressions and solving equations.

Curve Sketching

Curve sketching is the art of constructing a rough graph of a mathematical function or equation. Understanding the behavior of the function, such as periodicity, amplitude, and symmetry, can greatly help in visualizing the curve accurately.

For parametric curves, you should analyze how each individual parametric equation behaves. For instance, \( x = \cos t \) is periodic with a period of \( 2\pi \), and \( y = \cos 2t \) has a period of \( \pi \). Understanding these periodicities helps predict the curve's repetition and symmetry as it is sketched.

Additionally, knowing the ranges of each part of the parametric equations helps in placing points correctly. Once you have enough points, connect them smoothly to show the curve's overall behavior. Thus, curve sketching combines analytical skills with some creativity to produce an insightful visual representation of functions.