Problem 7 - Precalculus with Calculus Previews

Question

Sketch the curve that has the given set of parametric equations. $$ x=3 \cos t, y=5 \sin t, 0 \leq t \leq 2 \pi $$

Step-by-Step Solution

Answer

The curve is an ellipse centered at the origin, with semi-major axis 5 on the y-axis and semi-minor axis 3 on the x-axis.

Step 1: Identify the parametric equations

The exercise gives us the parametric equations: \( x = 3 \cos t \) and \( y = 5 \sin t \), where \( 0 \leq t \leq 2\pi \). These parametric equations define the x and y coordinates of points on the curve with respect to the parameter \( t \).

Step 2: Recognize the type of curve

The form of the parametric equations \( x = 3 \cos t \) and \( y = 5 \sin t \) resembles that of an ellipse. Comparing these with the standard parametric form \( x = a \cos t \) and \( y = b \sin t \), we see that these equations represent an ellipse with semi-major axis \( a = 3 \) and semi-minor axis \( b = 5 \).

Step 3: Eliminate the parameter

To verify the type of curve, we can eliminate the parameter \( t \). Using the trigonometric identities, square both equations: \( x^2 = (3 \cos t)^2 = 9 \cos^2 t \) and \( y^2 = (5 \sin t)^2 = 25 \sin^2 t \). Add these to obtain \( \frac{x^2}{9} + \frac{y^2}{25} = 1 \), which confirms the equation of an ellipse.

Step 4: Sketch the curve

Given the equation \( \frac{x^2}{9} + \frac{y^2}{25} = 1 \), the ellipse has its center at the origin. The semi-major axis is 5 units along the y-axis and the semi-minor axis is 3 units along the x-axis. Draw this ellipse, ensuring it extends 3 units along the x-axis and 5 units along the y-axis from the origin.

Key Concepts

Ellipse

An ellipse is a smooth, closed curve that resembles a stretched circle. It is characterized by its semi-major and semi-minor axes.
The semi-major axis is the longest diameter, while the semi-minor axis is the shortest.
In the context of parametric equations, an ellipse can be expressed using:

• \( x = a \cos t \)
• \( y = b \sin t \)

Here, \( a \) and \( b \) are the lengths of the semi-major and semi-minor axes, respectively.
The ellipse centered at the origin in the coordinate plane is symmetric about both the x-axis and y-axis.
When you plot these equations for \( 0 \leq t \leq 2\pi \), you get a complete ellipse. Explorers of conic sections often encounter ellipses because of their fascinating properties and occurrence in real-world scenarios, such as planetary orbits.

Trigonometric Identities

Trigonometric identities are fundamental tools in mathematics that relate the trigonometric functions to one another. These identities are crucial when working with parametric equations involving trigonometric functions.
For example, one of the most commonly used identities is

• \( \sin^2 t + \cos^2 t = 1 \)

This identity helps in eliminating the parameter 't' from parametric equations. In the case of an ellipse represented by parametric equations where \( x = a \cos t \) and \( y = b \sin t \), squaring these equations and using the identity simplifies the process to confirm the shape of the curve.
These identities simplify computations and provide a deeper understanding of the relationships between the trigonometric functions, making them indispensable in geometry and calculus.

Eliminate Parameter

Eliminating the parameter in parametric equations leads to a more familiar Cartesian equation. It's a way to show how the parametric curve relates to traditional x and y coordinates.
To eliminate a parameter like \( t \), you can use trigonometric identities as illustrated earlier.
In our example, starting from the equations \( x = 3 \cos t \) and \( y = 5 \sin t \), we square both equations:

• \( x^2 = 9 \cos^2 t \)
• \( y^2 = 25 \sin^2 t \)

Adding these gives \( \frac{x^2}{9} + \frac{y^2}{25} = 1 \), eliminating \( t \) and confirming the curve is an ellipse. This method not only verifies the nature of the curve but also provides a straightforward way to sketch it using familiar conic section equations.