Certainly, let's break down this question to simplify things.
Parametric equations are a set of equations that express the coordinates of the points of a curve, surface, etc., as functions of a parameter. In this case, our parameters are t (for time), x, and y.
Parametric curves are curves that are defined by parametric equations. Now, we are given the parametric equations as:
x = sin(pi*t)
y = cos(2*pi*t)
The definition range is from 0 to 0.5 for t values.
So let's sketch these equations step by step:
1. Start by defining t values from 0 to 0.5.
2. Next, we evaluate the x and y coordinates for these given t values using x = sin(pi*t) and y = cos(2*pi*t) respectively.
For example, for t=0, the x value will be sin(0)=0 and the y value will be cos(0)=1. For t=0.5, the x will be sin(pi/2)=1 and y will be cos(pi)= -1.
3. Then these calculated x and y values are plotted on a coordinate axis where x is horizontal and y is vertical.
4. Continue this process for all t values from 0 to 0.5. Each pair of (x,y) coordinates will form a point on the curve.
5. Finally, connect all points to form the curve described by the given parametric equations.
The curve defined by these parametric equations for 0 <= t <= 0.5 is actually a part of a circular shape starting from (0, 1) and ending at (1, -1) in the Cartesian coordinate system. This curve represents half rotation of point around the origin with a radius 1, counterclockwise, starting from the point (0,1) and ending at the point (1,-1).