Problem 14 - Algebra and Trigonometry

Question

\(3-24=A\) 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. $$ x=2 \cos t, \quad y=3 \sin t, \quad 0 \leq t \leq 2 \pi $$

Step-by-Step Solution

Answer

The curve is an ellipse with equation \(\frac{x^2}{4} + \frac{y^2}{9} = 1\).

Step 1: Understanding the Parametric Equations

We are given parametric equations: \(x = 2 \cos t\) and \(y = 3 \sin t\), with \(0 \leq t \leq 2\pi\). In these equations, the parameter \(t\) controls the values of \(x\) and \(y\) as \(t\) varies from 0 to \(2\pi\). We need to find out the shape of the curve represented by these parametric equations and also express it in terms of \(x\) and \(y\) only.

Step 2: Sketching the Curve

The given parametric equations represent an ellipse. The general form of a parametric equation for an ellipse is \(x = a \cos t\) and \(y = b \sin t\). Here, \(a = 2\) and \(b = 3\), indicating that we have an ellipse centered at the origin (0,0) with a horizontal radius (semi-major axis) of 2 units and a vertical radius (semi-minor axis) of 3 units. Sketch this ellipse by plotting points at key angles such as \(t = 0, \frac{\pi}{2}, \pi, \frac{3\pi}{2}\).

Step 3: Eliminating the Parameter

To eliminate the parameter \(t\), we want to express both \(x\) and \(y\) in terms of only one variable. From \(x = 2\cos t\), we have \(\cos t = \frac{x}{2}\), and from \(y = 3\sin t\), we have \(\sin t = \frac{y}{3}\). Using the trigonometric identity \(\cos^2 t + \sin^2 t = 1\), substitute to find: \(\left(\frac{x}{2}\right)^2 + \left(\frac{y}{3}\right)^2 = 1\).

Step 4: Simplifying the Rectangular Equation

Substitute the values into the trigonometric identity: \(\left(\frac{x}{2}\right)^2 + \left(\frac{y}{3}\right)^2 = 1\), which results in \(\frac{x^2}{4} + \frac{y^2}{9} = 1\). This is the rectangular coordinate equation for the ellipse.

Key Concepts

Rectangular Coordinate Equation

When working with curves described by parametric equations, translating them into a rectangular coordinate equation can make it easier to understand their shape and properties. In the given exercise, we have the parametric equations:

• \(x = 2 \cos t\)
• \(y = 3 \sin t\)

These equations define a curve in a 2D plane with the parameter \(t\) varying from 0 to \(2\pi\). The objective here is to eliminate \(t\) and express the relationship solely in terms of \(x\) and \(y\). Using trigonometric identities like \(\cos^2 t + \sin^2 t = 1\), we can combine these two parametric equations.
For this exercise, dividing the parametric equations results in:

• \(\cos t = \frac{x}{2}\)
• \(\sin t = \frac{y}{3}\)

Substituting these expressions into the identity \(\cos^2 t + \sin^2 t = 1\) leads us to: \[\left(\frac{x}{2}\right)^2 + \left(\frac{y}{3}\right)^2 = 1\] This is the rectangular coordinate equation of an ellipse.

Ellipse Equation

An ellipse is a geometric shape that appears as a stretched circle, having two focal points. The equation derived from parametric equations for an ellipse standard form is: \[\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\] where \(a\) is the semi-major axis, and \(b\) is the semi-minor axis.
In the exercise, the ellipse is centered at the origin with the parametric equations provided:

• \(x = 2 \cos t\)
• \(y = 3 \sin t\)

Here, \(a = 2\) and \(b = 3\). The parametric representation gives insight into how the ellipse is oriented and scaled on the axes. The transformation to the rectangular form \(\frac{x^2}{4} + \frac{y^2}{9} = 1\) confirms the vertical stretching due to \(b > a\).
Exploring the coordinate points using increments of \(t\) (like \(t = 0, \frac{\pi}{2}, \pi, \frac{3\pi}{2}\)) helps visualize how the ellipse traces over time onto the coordinate plane.

Trigonometric Identities

Trigonometric identities are crucial tools for manipulating and simplifying expressions, especially when dealing with periodic functions such as sine and cosine. These identities establish equality of different trigonometric expressions. One of the most fundamental identities is:

• The Pythagorean Identity: \(\cos^2 t + \sin^2 t = 1\)

This identity is pivotal in our exercise. It connects the squared cosine and sine values derived from the parametric equations \(x = 2 \cos t\) and \(y = 3 \sin t\), helping us convert them into a single equation in terms of \(x\) and \(y\) only.
Such identities help shorten the process of eliminating parameters and establish relationships between different parts of a trigonometric expression. Understanding and applying these identities not only simplify computations but also sharpen problem-solving skills in algebra and calculus.