We want to get this circle:
Start with the "usual" parameterization and mess with it until we find what we want. In the "usual" case, *x* goes from 1 to -1 then back to 1: draw circle and highlight where *x* is on *x* axis as drawing and *y* goes from 0 to 1, down to -1, then back up to 0: To draw the unit circle as specified in the problem, we want to have *x* go from 0 to 1, down to -1, then back to 0: and we can have *y* to go from 1 to -1 then back to 1: It looks like we want *x* to do what *y* does in the "usual" parameterization, and we want *y* to do what *x* does in the parameterization. Switch the equations: *x*(*t*) = sin t
*y*(*t*) = cos t
Use a graphing calculator to see what happens. We're tracing out the unit circle starting at the right place: The one thing we need to resolve is that the stylus is moving too slowly. We're supposed to travel all the way around the circle as *t* goes from 0 to π. Right now it's taking us from t = 0 to t = 2π to draw the whole circle. Cut the period of the equations in half, taking *x*(*t*) = sin (2t)
*y*(*t*) = cos (2t)
for 0 ≤ t ≤ π. Now we have what we want: To check the answer for a problem like this, stick the equations and the bounds for the parameter in a calculator. See if we find the correct graph, drawn in the correct direction. If the calculator draws the graph too quickly to see what direction it's going, play with the bounds for the parameter. If we graph the normal unit circle for 0 ≤ t≤ π and then for 0 ≤ t≤ 3π/2, we can see that the unit circle is traced in a counterclockwise direction. |