Question

Use technology (CAS or calculator) to sketch the parametric equations. $$ x=3 \cos t, \quad y=4 \sin t $$

Short answer

The parametric equations describe an ellipse centered at the origin with a semi-major axis of 4 and a semi-minor axis of 3.

Step-by-step solution

Understand the Parametric Equations

The given parametric equations are \( x = 3 \cos t \) and \( y = 4 \sin t \). These equations describe a curve in the coordinate plane as the parameter \( t \) varies.

Identify the Curve Type

The equations \( x = 3 \cos t \) and \( y = 4 \sin t \) suggest that the graph will be an ellipse. This is because \( x = 3 \cos t \) traces the horizontal motion and \( y = 4 \sin t \) traces the vertical motion.

Range of Parameter t

Typically, \( t \) ranges between \( 0 \) and \( 2\pi \) for a full cycle of these trigonometric functions. Within this range, the ellipse will be traced once completely.

Identify Major and Minor Axes

The coefficients in the equations indicate the semi-major and semi-minor axes' lengths. Here, 4 is the semi-major axis (since it appears with the \( \sin \) function, indicating vertical motion), and 3 is the semi-minor axis.

Use Technology to Sketch the Ellipse

Use a graphing calculator or computer algebra system (CAS) to plot the equations \( x = 3 \cos t \) and \( y = 4 \sin t \) over the interval \( 0 \leq t \leq 2\pi \). This will yield an ellipse centered at the origin with semi-major axis 4 and semi-minor axis 3.

Verify the Graph

Ensure that the resulting graph on your calculator or CAS matches an ellipse centered at the origin, extending 4 units vertically and 3 units horizontally.

Key concepts

Ellipse

An ellipse is a symmetrical shape that looks like a flattened circle. It's defined by two main components: the semi-major axis and the semi-minor axis. In simpler terms, think of an ellipse as an elongated circle, with its width and height potentially differing. This shape is common in nature and can be seen in many places, ranging from planetary orbits to the paths of comets.
A defining feature of an ellipse is its symmetry around two axes, providing it a unique, rounded but elongated form. When plotting an ellipse in terms of parametric equations, the position will change based on the values of the trigonometric functions related to the changing angle parameter, typically symbolized as \( t \). The shape of the ellipse is heavily influenced by the coefficients in these parametric equations.
Recognizing an ellipse from equations involves noting that both will involve trigonometric functions such as sine and cosine, setting the stage for cyclical, continuous paths traced out by the changing angle or time parameter.

Semi-Major Axis

The semi-major axis is the longest radius of an ellipse. It goes from the center to a point on the edge along the longest axis of the ellipse. In the given parametric equations, it's identified in the equation linked with the sine function. Specifically, in \( y = 4 \sin t \), the number 4 represents the semi-major axis. This means that the ellipse extends 4 units from the center vertically.
In geometrical terms, the length of the semi-major axis determines how 'stretched' or elongated the ellipse will appear. The greater this length, the more the ellipse will resemble a stretched circle.
Remember, an ellipse always has two semi-major axes. In reality, the axis isn't just about the number, but about symmetry and how it affects the overall aesthetics of the ellipse.

Semi-Minor Axis

The semi-minor axis is the shortest radius of an ellipse. It runs from the center to a point on the edge along the shortest axis of the ellipse. In the equations given, this is seen in the term \( x = 3 \cos t \). The number 3 signifies the length of the semi-minor axis, indicating the horizontal stretch of the ellipse.
The semi-minor axis is crucial because it determines the roundness of the ellipse. While the semi-major axis gives it elongation, the minor axis ensures the ellipse retains its smooth, rounded characteristics.
An ellipse's overall appearance is dictated by the ratio of its semi-major to semi-minor axes. A circle is a special type of ellipse where these lengths are equal, leading to no elongation in any direction.

Trigonometric Functions

Trigonometric functions, such as sine and cosine, play a pivotal role in defining parametric equations of an ellipse. These functions oscillate between -1 and 1, creating the paths that traces the shape of the ellipse as the parameter \( t \) varies from \( 0 \) to \( 2\pi \).
In parametric equations like \( x = 3 \cos t \) and \( y = 4 \sin t \), they determine the x and y coordinates for each angle \( t \). The cosine function affects the horizontal component while the sine function affects the vertical component.
By graphing these functions over the specified range, you map out the complete path of the ellipse. Understanding how sine and cosine oscillate is essential to predicting how and where the ellipse will be drawn in the coordinate plane.
Trigonometric principles not only help in plotting ellipses but are also fundamental in various applications like modeling waves, predicting planetary movements, and even in engineering fields.