91Ó°ÊÓ

Parametric equations are a way to express curves by finding both the x and y coordinates with the help of a parameter, usually represented by t.
Unlike regular functions, which write y directly in terms of x (y = f(x)), parametric equations provide two separate equations.
Each equation describes how x and y change with respect to the parameter t. For example, in a cycloid, the parametric equations are given by:
\( x = 2(t - \sin t) \) and \( y = 2(1 - \cos t) \).
These equations tell us the position of a point on the plane at any specific value of t.
This method is especially useful for describing more complex curves that are difficult to represent with a single y = f(x) function.
To understand how the cycloid behaves, we need to calculate the coordinates (x, y) for different values of t and then plot these points. For the cycloid, t ranges from 0 to 4Ï€.

Curve sketching is the process of drawing a graph based on a set of equations. With parametric equations, you generate a set of points by calculating x and y for a series of t values, then plotting these on a Cartesian plane.
For the cycloid, critical points to evaluate are when t equals 0, π, 2π, 3π, and 4π. Calculating these:

• When t = 0: (x, y) = (0, 0)
• When t = Ï€: (x, y) = (2Ï€, 4)
• When t = 2Ï€: (x, y) = (4Ï€, 0)
• Continue similarly for t = 3Ï€ and 4Ï€.

Plot these points on a Cartesian plane.
Join the points smoothly to identify the cycloid shape. This step-by-step plotting is essential to visualize any complex curve.

Trigonometric functions like sine and cosine play a crucial role in defining the parametric equations for a cycloid.
Sine, \sin(t)\, and cosine, \cos(t)\, are periodic functions that oscillate between -1 and 1.
Here's how they help in this context:

• \( x = 2(t - \sin t) \): The term (t - \sin t) defines the horizontal displacement and varies as sine function oscillates.
• \( y = 2(1 - \cos t) \): The term (1 - \cos t) defines the vertical displacement and varies as cosine function oscillates.

These oscillations create the distinctive arches of the cycloid as the points are plotted.
Understanding how these trigonometric functions interact for different values of t (0 ≤ t ≤ 4π) helps us predict the shape and behavior of the cycloid.