Question Sheet 4 - Challenge Problem

The equation for a circle is:

				          	 (r^2)=(x^2)+(y^2)                                  				     	      (1)
  
It can be expressed using parametric equations, i.e.
  
				     x(t) = r cos(t), y(t) = r sin(t).             			           	      (2)
				
In equation (1), we do not have an explicit form for x or y. If we rearrange equation (1) we can get an expression for x and y as follows:

				x(y)=(Sqrt[(r^2)-(y^2)]), y(x)=(Sqrt[(r^2)-(y^2)])        			    		     (1b)

However we note that the equation x(y) depends on y, and the equation y(x) depends on x, so we do not have functions that are dependent on independent variables. This is the motivation for introducing parametric equations, since here, t is an independent variable. In physics we often want to plot ellipses (a circle is a special case of an ellipse), for example, the orbit of planets or the orbit of a charged particle in a field. Typically it is useful to find these in terms of parametric equations. Thus it is useful to be able to plot these.

Now that the help file has been introduced, there is much more freedom to explore Mathematica. Search for ParametricPlot in the help file.

1. What is the form of the ParametricPlot command?

2. Assign a value of 2 to r. Use text in your code to inform the reader that r is the radius of a circle.

3. Use ParametricPlot to plot a circle. Label the x- and y-axis with x(t) and y(t) respectively. Also, label your graph with a suitable title.

[Hint: It is often worthwhile plotting your graph, and then going back and adding your labels once you know the code is working.]