91Ó°ÊÓ

Parametric equations are a powerful way to describe curves in mathematics, using parameters. Here, the position of a point is defined with respect to a variable, commonly an angle or time. In our given example, the point \(P\) has its position represented by the vector \(\vec{r} = a \cos \theta \hat{i} + b \sin \theta \hat{j}\). This setup uses \(\theta\) as the parameter. The parametric form allows visualizing how a point moves across a plane as \(\theta\) changes.
In this case, the coordinates \(x\) and \(y\) are derived from trigonometric functions of \(\theta\), giving us:

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

Each parameterization captures the path in a more dynamic way, ideal for problems involving motion or rotation. By varying \(\theta\), we explore all possible positions \(P\) can occupy on its trajectory.

To move from parametric equations to the standard form of a curve, trigonometric identities come to the rescue. These identities are foundational tools that relate the angles and sides of triangles. The most important identity used here is the Pythagorean identity: \[\cos^2 \theta + \sin^2 \theta = 1\]In the context of our problem, it helps to eliminate the parameter \(\theta\) when translating \(x\) and \(y\):
Start by expressing \(\cos \theta\) and \(\sin \theta\) in terms of \(x\) and \(y\):

• \(\cos \theta = \frac{x}{a}\)
• \(\sin \theta = \frac{y}{b}\)

Placing these into the Pythagorean identity, we derive:
\[\left(\frac{x}{a}\right)^2 + \left(\frac{y}{b}\right)^2 = 1\]
This final equation matches the standard ellipse equation, reinforcing the critical role of trigonometric identities in mathematical transformations.

The behavior of a particle moving through space can be visualized through its path, an essential concept in physics and engineering. The specific path of the particle in this exercise is revealed through the relationship of its coordinates over the parameter \(\theta\). Given the positions \(x = a \cos \theta\) and \(y = b \sin \theta\), as \(\theta\) varies, it forms an ellipse on the Cartesian plane.
The rate of change of \(\theta\), denoted by \(\omega\), plays a significant role in defining the speed of this particle. While we don't focus on the mechanics of motion here, knowing \(\omega\) helps conceptualize how quickly or slowly the particle traces its elliptical path.
Understanding the particle path by its equation \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\) provides deeper insights into the nature of its movement. It identifies both the geometric shape and how different parameters affect the journey across it. This conceptual clarity is crucial for solving complex real-world problems across various domains.