91Ó°ÊÓ

When working with parametric equations, one of the main tasks is converting them into a Cartesian equation. This process involves eliminating the parameter—in this case, "t" from the equations.
In the given exercise, we started with the parametric equations:

• \( x = \sin t \)
• \( y = 1 - \cos t \)

To find the Cartesian equation, we make use of trigonometric identities to remove "t."
Squaring both equations gives us:

• \( x^2 = \sin^2 t \)
• \( (1 - y)^2 = \cos^2 t \)

Using the identity \( \sin^2 t + \cos^2 t = 1 \), we substitute and simplify to obtain a relationship just in terms of \( x \) and \( y \):\[ x^2 + (1-y)^2 = 1 \]
This results in the final Cartesian equation that describes the curve in a way that's free of the parameter \( t \).

Sketching curves described by parametric equations helps visualize how the variables change with respect to the parameter "t." In this exercise, plotting specific points obtained by substituting values of "t" is crucial.
By choosing values like \( t = 0, \frac{\pi}{2}, \pi, \frac{3\pi}{2}, \) and \( 2\pi \), we found points like (0,0), (1,2), (0,1), (-1,2), and back to (0,0).
Connecting these points smoothly, a loop-like curve emerges. The direction in which the curve is traced is indicated with an arrow from \( t=0 \) to \( t=2\pi \), moving counterclockwise.
This visualization helps understand not just the shape and nature of the curve but also how the curve is traversed over a cycle of the parameter. Curve sketching can provide a clear geometric interpretation of parametric paths.

Trigonometric identities are powerful tools in converting parametric equations to their Cartesian forms. They simplify the process of removing the parameter.
Here, identities such as \( \sin^2 t + \cos^2 t = 1 \) provide a connection between the squared terms gained from the parametric equations.
By squaring each part of our parametric equations and utilizing this core trigonometric identity, we are able to eliminate "t."
This results in the equation: \[ x^2 + (1-y)^2 = 1 \]
Trigonometric identities not only help in mathematical transformations but also assist in understanding relationships between angles and sides within trigonometric functions. Access to such identities is essential for anyone working with angles and periodic patterns in mathematics.

Completing the square is a crucial algebraic technique used to transform quadratic expressions into a different form, often to reveal geometric properties of an equation.
In the Cartesian equation derived:\[ x^2 + y^2 - 2y = 0 \]
Completing the square for the "y" part involves:

• Rewriting \( y^2 - 2y \) as \( (y-1)^2 - 1 \)

This transformation helps in rearranging the equation into:\[ x^2 + (y - 1)^2 = 1 \]
This reveals a standard circle equation centered at (0,1) with radius 1 on the Cartesian plane. Completing the square offers a technique to understand forms, centers, and dimensions of these geometric figures, allowing mathematical equations to be interpreted as real-world shapes.