91Ó°ÊÓ

In mathematics, a parametric equation expresses a set of quantities as explicit functions of independent parameters. Often, we use parameters such as \( t \) to describe the path of a point in space over time. However, sometimes we want to eliminate the parameter to find a direct relationship between two variables, like \( x \) and \( y \).

To eliminate a parameter, follow these steps:

• Resolve the equations to express the parameter explicitly in terms of another variable, if possible.
• Equate the expressions for the parameter from the separate equations.
• Simplify to find a direct relationship between the variables, removing the parameter.

Using the exercise example, we started with \( x = 3 \sin t \) and \( y = 2 \sin t \). By expressing \( \sin t \) in terms of \( x \) and \( y \), and equating as shown, we derive a direct equation \( y = \frac{2}{3}x \) without involving the parameter \( t \). This process is widely used in algebra and trigonometry for solving equations and visualizing curves without focusing on the path or direction.

Now that we have eliminated the parameter, we are left with a direct relationship between \( x \) and \( y \): \( y = \frac{2}{3}x \). This equation represents a straight line on the coordinate plane, revealing how the two variables change in relation to one another.

In essence, the relationship shows:

• For every unit increase in \( x \), \( y \) increases by \( \frac{2}{3} \).
• The line passes through the origin (0,0) because when \( x = 0 \), \( y \) is also 0.
• The slope of the line is \( \frac{2}{3} \), indicating its steepness.

Understanding this relationship is crucial in interpreting how one variable changes as the other changes. It reveals the linear correlation between \( x \) and \( y \) without the parameter, simplifying complex paths to straightforward equations.

Parametric equations often involve trigonometric functions, as in this exercise where \( \sin t \) is used. Trigonometric equations such as \( \sin, \cos, \, \text{and} \, \tan \) are vital tools in modeling periodic phenomena like waves or oscillations, because they offer:

• An understanding of periodicity and circular motion due to their nature on the unit circle.
• Insight into amplitude, frequency, and phase differences in various applications.
• Simplification of complex variables into manageable trigonometric relationships as shown in the exercise by expressing \( \sin t \) in terms of \( x \) and \( y \).

By equating the trigonometric parts from each parametric expression, we transform these relationships into simple linear or curve equations. In our exercise, trigonometric conversion allowed us to disregard \( t \) entirely and find a clean, easy-to-understand relationship between \( x \) and \( y \). This approach is particularly useful in engineering, physics, and computer graphics, where modeling movement and waves is essential.