The following applet can be used to investigate curves drawn in polar coordinates. Simply enter the function r(t) and the values a and b (in radians). The curve r(t) is drawn in the usual x,y-plane in the main window. The curve r(t) is drawn in the t,r(t)-plane in the pulldown window.

The values a and b can be changed by simply typing a new value, such as "1.2345", "pi/2", "sqrt(5)+cos(3)", etc. You may also change these values by using the up/down arrow keys or dragging the corresponding point left or right. To move the center of the graph, simply drag any point to a new location. To label the x-axis in radians (i.e. multiples of pi), click on the graph and press "control-r". To switch back, simply press "control-r" again.

Here is a list of functions that can be used with this applet.

Use the above applet to:

• investigate the polar curve r(t) = A cos(t) + B sin(t) for various values of A and B. What kind of geometric object is it? Describe this curve in Cartesian coordinates.
• investigate the number of loops in the curve r(t) = sin(kt) for k=1,2,3,4,....