91Ó°ÊÓ

The Pythagorean identity is a fundamental concept in trigonometry. It states that for any angle \( t \), the equation \( \sin^2 t + \cos^2 t = 1 \) holds true. This identity is derived from the Pythagorean theorem and is crucial when working with trigonometric equations. In the context of parametric equations, it provides a bridge to eliminate the parameter \( t \) and link equations involving sine and cosine directly.

For instance, if you have \( \cos t = \frac{x - 2}{5} \) and \( \sin t = \frac{y - 7}{5} \), you can apply the Pythagorean identity to find that:

• \( \left( \frac{x - 2}{5} \right)^2 + \left( \frac{y - 7}{5} \right)^2 = 1 \)

This conversion simplifies the parametric form to an algebraic one, setting the stage for identifying the geometric nature of the path or curve.

Circle equations in a two-dimensional plane are typically written in the form \( (x - h)^2 + (y - k)^2 = r^2 \), where \((h, k)\) represents the center and \( r \) the radius of the circle.

In solving parametric equations, transforming them into this standard circle equation is often the goal, as it offers a clear geometric understanding of the path. For example, by substituting trigonometric expressions found in parametric equations into the Pythagorean identity, you might end up with:

• \( \frac{(x - 2)^2}{25} + \frac{(y - 7)^2}{25} = 1 \)

Multiplying out the fractions yields:

• \( (x - 2)^2 + (y - 7)^2 = 25 \)

This equation confirms that the curve is a circle with a center of \((2, 7)\) and a radius of \( 5 \). Understanding this result helps in interpreting the motion of the particle along a circular path.

Eliminating parameters in equations is a powerful tool for simplifying expressions and uncovering underlying geometric relationships. In parametric equations, \( t \) is often the parameter, a variable representing time or another continuous value that alters the values of \( x \) and \( y \).

The objective is to express \( x \) and \( y \) directly in relation to each other, without \( t \). By isolating \( \cos t \) and \( \sin t \) in the original parametric equations and substituting them into a relationship like the Pythagorean identity, the parameter \( t \) is effectively removed:

• \( \cos t = \frac{x - 2}{5} \)
• \( \sin t = \frac{y - 7}{5} \)
• Apply identity: \( \left(\frac{x - 2}{5}\right)^2 + \left(\frac{y - 7}{5}\right)^2 = 1\)

This process simplifies the representation of the curve or path, allowing a more straightforward analysis of its properties, such as its shape or center.