Problem 13 - Precalculus

Question

Graph the parametric equations after eliminating the parameter t. Specify the direction on the curve corresponding to increasing values of \(t\). \(t\) is \(0 \leq t \leq 2 \pi\). $$x=2 \sin t, y=3 \cos t$$

Step-by-Step Solution

Answer

The curve is an ellipse \(\frac{x^2}{4} + \frac{y^2}{9} = 1\), traced counter-clockwise.

Step 1: Understand the Problem

We need to graph parametric equations given by \(x = 2 \sin t\) and \(y = 3 \cos t\). We will first eliminate the parameter \(t\) to find a relationship between \(x\) and \(y\). We will then determine the direction of the curve for increasing values of \(t\).

Step 2: Use Trigonometric Identities

We express \(\sin t\) and \(\cos t\) in terms of \(x\) and \(y\): \(\sin t = \frac{x}{2}\) and \(\cos t = \frac{y}{3}\). Since \(\sin^2 t + \cos^2 t = 1\), we substitute these into the identity to eliminate \(t\).

Step 3: Simplify the Equation

Using the identities from Step 2, we substitute into the Pythagorean identity: \(\left(\frac{x}{2}\right)^2 + \left(\frac{y}{3}\right)^2 = 1\). Simplify it to find the equation of the curve: \(\frac{x^2}{4} + \frac{y^2}{9} = 1\).

Step 4: Identify the Shape and Direction

The equation \(\frac{x^2}{4} + \frac{y^2}{9} = 1\) represents an ellipse centered at the origin. Next, we need to determine the direction of the curve. For increasing \(t\), we look at changes in positions for specific values such as \(t = 0\), \(t = \frac{\pi}{2}\), \(t = \pi\), \(t = \frac{3\pi}{2}\), and \(t = 2\pi\). We find that the curve is traced counter-clockwise.

Key Concepts

Eliminating Parameters

Eliminating parameters in a set of parametric equations allows us to simplify the problem by reducing it from a system based on a third variable, such as time, to a single equation in two variables. For the parametric equations \(x = 2 \sin t\) and \(y = 3 \cos t\), we can eliminate the parameter \(t\). Using trigonometric identities, we express \(\sin t\) and \(\cos t\) in terms of \(x\) and \(y\). This is accomplished by solving for \(\sin t\) as \(\frac{x}{2}\) and \(\cos t\) as \(\frac{y}{3}\). By inserting these expressions into the Pythagorean identity \(\sin^2 t + \cos^2 t = 1\), we effectively remove the dependence on \(t\), achieving a simple Cartesian relationship:

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

This brings us to an equation that we can graph in the \(xy\) plane, without having to consider the parameter anymore.

Ellipse

The resulting equation \(\frac{x^2}{4} + \frac{y^2}{9} = 1\) is recognized as the standard form of an ellipse. An ellipse is a geometric shape and it looks like a stretched circle. It is defined mathematically by such an equation, where the denominators determine the extent of stretching along the x or y axes.

• The numerator \(x^2/4\) shows that the ellipse is stretched along the x-axis with a semi-axis length of 2.
• Similarly, \(y^2/9\) shows that the ellipse is stretched along the y-axis with a semi-axis length of 3.

Consequently, this gives us information about the ellipse’s center, which in this case is at the origin \(0,0\). Understanding the positioning and shape of an ellipse is important for geometry and physics, where elliptical motion is common.

Trigonometric Identities

Trigonometric identities, like the Pythagorean identity, are foundational tools in math. They help relate different trigonometric functions and allow us to manipulate and transform expressions. The identity \(\sin^2 t + \cos^2 t = 1\) is particularly useful for converting parametric equations into Cartesian form. In our earlier steps, we used this identity to replace \(t\) with expressions involving \(x\) and \(y\), converting a parametric equation into a standard geometric shape.

• This identity permits transitions between the circular path of \(\sin\) and \(\cos\) into a recognizable Cartesian curve, such as an ellipse.
• Understanding these identities helps deepen comprehension of how different kinds of geometric transformations occur using trigonometry.

By mastering trigonometric identities, students are equipped to handle versatile mathematical problems with greater confidence.

Graphing Curves

Graphing curves derived from parametric equations is an engaging activity that brings mathematical expressions to life. By plotting points derived from parametric definitions and observing their evolution as the parameter changes, we can visualize the path an object follows in space. Steps for effectively graphing curves involve:

• Eliminating parameters to find a Cartesian equation.
• Identifying the shape of the curve, like the ellipse here.
• Determining the direction of traversal by examining points at critical values of the parameter.

For the current example, as \(t\) progresses from 0 to \(2\pi\), the curve traced out is counter-clockwise. Studying how these curves work serves crucial functions in physics and engineering, where understanding the motion and characteristics of circular and elliptical paths aid critical analysis and design processes.