91Ó°ÊÓ

Trigonometric identities play a fundamental role in simplifying complex equations and expressions involving angles and sides of triangles. Here, the identity \( \sin^2 t + \cos^2 t = 1 \) is used effectively to eliminate the parameter \( t \) from the parametric equations. This identity is derived from the Pythagorean theorem and is crucial in converting trigonometric expressions to a form that is easier to work with.

• This identity states that the square of the sine of an angle plus the square of the cosine of that angle is always equal to 1.
• It helps in expressing one trigonometric function in terms of another, as shown in the given solution where \( \sin^2 t \) and \( \cos^2 t \) are in terms of \( x \) and \( y \).

The ability to apply such identities is essential in solving problems where parameters need to be eliminated. By substituting these expressions back into the identity, we are able to derive a relationship solely between \( x \) and \( y \). This simplifies the understanding and visualization of the problem.

Cartesian coordinates provide a way to describe points on a two-dimensional plane using an \( x \) and \( y \) axis. In this exercise, the parametric equations \( x = 2\sin^2 t \) and \( y = 3\cos^2 t \) are transformed into a single Cartesian equation.

• The conversion process involves substituting \( \sin^2 t \) and \( \cos^2 t \) with expressions in terms of \( x \) and \( y \).
• Here, \( \sin^2 t = \frac{x}{2} \) and \( \cos^2 t = \frac{y}{3} \) are utilized to formulate the equation \( \frac{x}{2} + \frac{y}{3} = 1 \).

By eliminating the parameter \( t \), the result is a straightforward linear equation \( 3x + 2y = 6 \), which represents the curve in the Cartesian plane. This method allows students to understand the link between parametric equations and their Cartesian counterparts.

Curve sketching is a powerful tool in visualizing mathematical functions and their behaviors on a graph. With the Cartesian equation \( 3x + 2y = 6 \) derived from the parametric forms, sketching the curve becomes more manageable. Since this equation corresponds to a straight line, the task of sketching simplifies to plotting this line on the Cartesian plane.

• The line can easily be sketched by determining its intercepts: when \( x = 0 \), \( y = 3 \) and when \( y = 0 \), \( x = 2 \).
• Connecting these intercepts gives the complete representation of the line.

Understanding the direction of increasing \( t \) adds depth to the sketching task. As \( t \) increases, \( x \) and \( y \) values change, indicating movement along the line from one intercept to the other. This concept aids in better understanding the dynamics of the function or curve being analyzed.