91Ó°ÊÓ

Eliminating parameters in parametric equations is a crucial process in mathematics. It helps convert parametric equations into a single, clean expression that describes a curve. When dealing with parametric equations, like those given in the exercise:

• \( x = 2 \cos t \)
• \( y = 5 \sin t \),

the variables \( x \) and \( y \) depend on a parameter, \( t \).
By eliminating the parameter, we focus solely on the relationship between \( x \) and \( y \), removing their dependence on \( t \). This simplifies the process of analyzing and graphing the curve.
To eliminate \( t \), we utilize relationships or identities that link \( x \) and \( y \). This exercise uses the trigonometric identity \( \cos^2 t + \sin^2 t = 1 \), making it easier to replace \( \cos t \) and \( \sin t \) with terms involving \( x \) and \( y \). This results in deriving a direct equation between \( x \) and \( y \) without the parameter \( t \).

The equation of an ellipse can be beautifully unveiled through the process of parameter elimination. When the parameters are successfully removed from the parametric equations, we end up with the standard form of the ellipse equation:

• \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \)

In this standard form, \( a \) and \( b \) represent the semi-major and semi-minor axes of the ellipse respectively.
By employing our simplified equation \( \frac{x^2}{4} + \frac{y^2}{25} = 1 \), it's evident that the ellipse is oriented along both the x-axis and y-axis. The values 4 and 25 at the denominators relate to the squares of the lengths of the semi-axes:

• The semi-major axis, \( b = 5 \).
• The semi-minor axis, \( a = 2 \).

This equation describes an ellipse, which is crucial for graphing the curve.
Identifying this key form helps in sketching and understanding the nature of the path described by the equation.

Trigonometric identities are incredibly useful tools in mathematics, especially when dealing with parametric equations. These identities act as bridges, connecting different trigonometric functions to transform or simplify equations.
In the exercise, the identity \( \cos^2 t + \sin^2 t = 1 \) is pivotal. This specific identity is fondly known as the Pythagorean identity and is primarily useful for eliminating parameters from equations.
Here's how it works:

• Given \( x = 2 \cos t \) and \( y = 5 \sin t \), you express \( \cos t \) and \( \sin t \) in terms of \( x \) and \( y \) respectively, as \( \cos t = \frac{x}{2} \) and \( \sin t = \frac{y}{5} \).
• Substituting these into \( \cos^2 t + \sin^2 t = 1 \), you get: \[ \left(\frac{x}{2}\right)^2 + \left(\frac{y}{5}\right)^2 = 1 \]

This identity effectively removes \( t \), leaving an equation that directly links \( x \) and \( y \).
Mastering such identities not only aids in equation manipulation but also enhances one's ability to solve more complex trigonometric problems.