Problem 24 - 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. $$\begin{aligned} &x=2 \cos \theta\\\ &y=3 \sin \theta \end{aligned}$$

Step-by-Step Solution

Answer

The sketch of the parametric curve is an ellipse with its major axis along the y-axis, and oriented in a counterclockwise direction. When the parameter \( \theta \) is eliminated, the rectangular equation representing the curve is \((x^2 / 4) + (y^2 / 9) = 1\), representing the same ellipse.

Step 1: Sketching the parametric curve

To sketch the curve, for each value of \( \theta \) from 0 to 2\( \pi \), calculate the corresponding x and y values using the given parametric equations. Mark these points on the coordinate plane. When \( \theta = 0 \), we have \( x = 2 \) and \( y = 0 \). For \( \theta = \pi / 2 \), \( x = 0 \), \( y = 3 \). For \( \theta = \pi \), \( x = -2 \) and \( y = 0 \).And for \( \theta = 3\pi / 2 \), \( x = 0 \), \( y = -3 \). Connect these points to visualize the curve. Notice a cycle repeats as \( \theta \) goes from 0 to 2\( \pi \), thus indicating the orientation of the curve.

Step 2: Convert to the rectangular form

In order to eliminate the parameter \( \theta \)and write the rectangular equation, utilize the fact that \( \sin ^{2} \theta +\cos ^{2} \theta = 1 \). Squaring both equations, we get \( x^2=4 \cos^{2} \theta \) and \( y^2=9 \sin^{2} \theta \). Dividing the first equation by 4 and the second by 9, then adding the results, this yields \((x^2 / 4) + (y^2/ 9) = 1\), a standard equation of an ellipse in the rectangular form.

Key Concepts

Rectangular Equation

A rectangular equation is an expression that describes a curve using Cartesian coordinates, typically denoted as \(x\) and \(y\). It is derived from parametric equations by eliminating the parameter, such as \(\theta\). In our case, we started with the parametric equations \(x = 2 \cos \theta\) and \(y = 3 \sin \theta\). To convert this to a rectangular equation, we use a known trigonometric identity: \(\sin^2 \theta + \cos^2 \theta = 1\). By squaring and rearranging the parametric equations, we obtained an expression in terms of \(x\) and \(y\):

• \(x^2 = 4 \cos^2 \theta\)
• \(y^2 = 9 \sin^2 \theta\)

Divide these equations respectively by 4 and 9, and sum them up, giving us \(\frac{x^2}{4} + \frac{y^2}{9} = 1\), the rectangular form of an ellipse. Notice how these operations effectively eliminate the parameter \(\theta\), enabling us to graph the ellipse directly in the coordinate plane without calculating specific points for \(\theta\). When interpreting this equation, ensure to consider the domain—since it originates from trigonometric functions, typically \(0 \leq \theta \leq 2\pi\).

Ellipse

An ellipse is a type of conic section that appears as a stretched circle. Its mathematical equation in standard form is \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\), where \(a\) and \(b\) represent the semi-axes lengths. In our example, \(a = 2\) and \(b = 3\), indicating an ellipse stretched more along the \(y\)-axis.

• The semi-major axis is the longest diameter of the ellipse. In this case, it is in the direction of the \(y\)-axis.
• The semi-minor axis is perpendicular to the major axis, and in this ellipse, it is along the \(x\)-axis.

The orientation of an ellipse is deduced from the axis lengths and their respective directions in the coordinate plane.Ellipses are central to many fields, with applications spanning astronomy to engineering. They're particularly significant in orbits, as many celestial bodies follow elliptical paths.

Coordinate Plane

The coordinate plane, also known as the Cartesian plane, is a two-dimensional plane defined by the \(x\) and \(y\) axes intersecting at a right angle at the origin point \((0,0)\). This plane allows for the plotting and understanding of curves, lines, and any other geometrical shapes.

• The \(x\)-axis is the horizontal line; measurements increase towards the right and decrease towards the left.
• The \(y\)-axis is the vertical line; measurements increase upward and decrease downward.

Understanding the coordinate plane is fundamental when graphing parametric equations, like our example with \(x = 2 \cos \theta\) and \(y = 3 \sin \theta\). By marking points based on these values, the orientation and shape of the described curve, an ellipse, become apparent once they are plotted together.

Trigonometric Identities

Trigonometric identities are mathematical equations that relate the trigonometric functions \(\sin\), \(\cos\), and \(\tan\) to one another. A well-known identity useful in our example is \(\sin^2 \theta + \cos^2 \theta = 1\). This identity stems from the Pythagorean Theorem, considering a unit circle where the hypotenuse is 1.These identities are powerful tools for simplifying expressions, converting between parametric and rectangular equations, and solving problems within trigonometry. They transform complex relationships into more accessible forms—as when transforming our parametric equations \(x = 2 \cos \theta\) and \(y = 3 \sin \theta\) into the rectangular equation \(\frac{x^2}{4} + \frac{y^2}{9} = 1\).When working in math, especially with trigonometric components, these identities simplify calculations and are essential for deeper mathematical exploration.