91Ó°ÊÓ

In trigonometry, identities like \( \sin^2 t + \cos^2 t = 1 \) are foundational relationships that hold for all angles. Such identities are incredibly useful in converting and simplifying equations, especially when dealing with parametric equations. In the given exercise, knowing this identity allowed us to express \( \cos^2 t \) in terms of \( \sin^2 t \). This is critical because it helps us eliminate the parameter \( t \) from the equations by relating the expressions for \( x \) and \( y \) through the identity. By substituting \( \cos^2 t = 1 - \sin^2 t \) into the \( y \) equation, we gain a common ground to interconnect our expressions in a straightforward manner. This simplification is key for the elimination process, which we'll cover next.

The task of eliminating parameters in parametric equations involves expressing the relationship directly between the x and y variables, without referencing the parameter \( t \). This process often starts by using trigonometric identities to express one part of the equation in terms of another common element. In our exercise, by substituting \( \sin^2 t = x/2 \) into the equation for \( y \), we derive a direct relation: \( y = 3 - \frac{3}{2}x \). This transformation allows us to eliminate the parameter \( t \), unveiling the path of the curve in terms of \( x \) and \( y \) alone. This new equation represents a straight line, making further analysis and sketching much simpler and more intuitive.

Curve sketching involves understanding the behavior of a curve as influenced by the parameter \( t \). In our scenario, once we've related \( x \) and \( y \) directly through \( y = 3 - \frac{3}{2}x \), sketching becomes straightforward. It's key to note the direction of the curve as \( t \) changes. Here, as \( t \) moves from 0 to \( \pi/2 \), \( \sin^2 t \) increases, causing \( x \) to increase from 0 to 2 and \( y \) to decrease from 3 to 0. This indicates that the line is traversed from right to left. Noticing this direction is important since it informs us of how the curve evolves over the interval of \( t \). Such sketches help visualize the problem and enhance understanding of the overall transformation.