Problem 12 - 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} $$ x=2 \cos t, \quad y=3 \sin t, \quad 0 \leq t \leq 2 \pi $$

Step-by-Step Solution

Answer

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

Step 1: Understand the Parametric Equations

We are given the parametric equations: \(x = 2\cos t\) and \(y = 3\sin t\). These equations define the coordinates \(x\) and \(y\) in terms of the parameter \(t\). The range for the parameter \(t\) is from \(0\) to \(2\pi\).

Step 2: Analyze the Range of the Parameter \(t\)

The parameter \(t\) varies from \(0\) to \(2\pi\), which means it completes one full cycle for the trigonometric functions \(\cos t\) and \(\sin t\), describing an entire ellipse.

Step 3: Sketch the Curve

Substitute different values of \(t\) within its range into the parametric equations to find corresponding points \((x, y)\) and plot them. Common values include \(t = 0, \frac{\pi}{2}, \pi, \frac{3\pi}{2}, 2\pi\) which give us points (2, 0), (0, 3), (-2, 0), (0, -3), (2, 0). These points outline an ellipse centered at the origin with semi-major axis length 3 (along y-axis) and semi-minor axis length 2 (along x-axis).

Step 4: Eliminate the Parameter (Find Rectangular Equation)

To eliminate \(t\), use the identities \(\cos^2 t + \sin^2 t = 1\). Solve \(x = 2\cos t\) for \(\cos t\): \(\cos t = \frac{x}{2}\). Solve \(y = 3\sin t\) for \(\sin t\): \(\sin t = \frac{y}{3}\). Substitute these into the identity: \(\left(\frac{x}{2}\right)^2 + \left(\frac{y}{3}\right)^2 = 1\). Simplifying gives: \(\frac{x^2}{4} + \frac{y^2}{9} = 1\), which is the equation of an ellipse.

Key Concepts

Ellipse

An ellipse is a smooth, closed curve that resembles an elongated circle. It is defined mathematically as the set of all points for which the sum of the distances to two fixed points (called foci) is constant. In the context of our parametric equations, an ellipse occurs when the equations define a path that loops around in a symmetric, oval shape.
When you plot the points generated by the parametric equations \(x = 2\cos t\) and \(y = 3\sin t\), the resulting shape is an ellipse. Here, the longest diameter (known as the major axis) runs along the \(y\)-axis with a length of 6, while the shortest diameter (the minor axis) goes along the \(x\)-axis with a length of 4. This aligns with the ellipse's definition and structure, forming a symmetrical path around the origin.

• The semi-major axis (half of the major axis) has length 3.
• The semi-minor axis (half of the minor axis) has length 2.

Overall, ellipses are a vital concept in various mathematical and physical contexts due to their unique geometric and algebraic properties.

Rectangular-Coordinate Equation

A rectangular-coordinate equation is a mathematical expression that relates coordinates in a Cartesian plane. In the case of an ellipse, the rectangular-coordinate equation provides a way to describe the ellipse without involving a parameter like \(t\).
To convert from parametric to rectangular form, we eliminate the parameter \(t\). For our problem, given \(x = 2\cos t\) and \(y = 3\sin t\), we use trigonometric identities to find a direct relation between \(x\) and \(y\).
We solve for \(\cos t\) and \(\sin t\):

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

We then apply the Pythagorean identity \(\cos^2 t + \sin^2 t = 1\) and substitute:

• \[\left(\frac{x}{2}\right)^2 + \left(\frac{y}{3}\right)^2 = 1\]

Simplify this to obtain:

• \[\frac{x^2}{4} + \frac{y^2}{9} = 1\]

This equation is the rectangular-coordinate equation for the ellipse. It clearly demonstrates how \(x\) and \(y\) are related, free from any parametric variable.

Trigonometric Functions

Trigonometric functions are fundamental in describing periodic phenomena. They relate angles of a right triangle to the ratios of the triangle’s side lengths. In parametric equations, these functions help represent circular and elliptical paths.
In our exercise, the functions \(\cos t\) and \(\sin t\) are used. These functions vary in the interval from \(0\) to \(2\pi\), which corresponds to one complete revolution around a unit circle.

• The cosine function \(\cos t\) describes the horizontal component of the circular motion, affecting the \(x\)-coordinate.
• The sine function \(\sin t\) corresponds to the vertical component, affecting the \(y\)-coordinate.

In our parametric equations, \(x = 2\cos t\) and \(y = 3\sin t\):

• \(\cos t\) gives the ratio for \(x\) with a stretch factor of 2, making the path 4 units wide.
• \(\sin t\) gives the ratio for \(y\) with a stretch factor of 3, making it 6 units tall.

This illustrates how trigonometric functions transform circular motion into an elliptical path, showcasing their utility in modeling diverse motions and shapes.