Parametrizations of the Unit Circle Examples

Give a parameterization of the unit circle that starts at the point (1, 0) and draws the unit circle once in a clockwise direction for 0 ≤ t ≤ 2π.

With the labels and arrows, we're trying to find this graph:

We want the etch-a-sketch stylus to start at the same point and travel the same speed as usual, but we want it to travel the opposite direction. What does that mean?

When we don't want the stylus to be at . We want it to be at .

When we don't want the stylus to be at (0, 1). We want it to be at (0, -1).

To summarize, we want the y-coordinates to be negative, and the x-coordinates to remain unchanged.

We want to stick a negative sign in front of the y equation, and leave the equation for x unchanged. Our new parameterization is

x(t) = cos t
y(t) = -sin t

for 0 ≤ t ≤ 2π.

Give parametric equations and bounds for the parameter that traces the unit circle clockwise so that the etch-a-sketch stylus is at (1, 0) when t = 0 and again when t = π. This is a sausage pizza because it's made from the same stuff as pepperoni but tastes different.

We want to get this circle:

We'll 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.

To draw the unit circle as specified in the problem, we want to have x go from 1 to 0, over to -1, then back to 0.

and we can have y  go from 0 to -1 then back to 0, and up to 1.

It looks like we want x to do what x does in the "usual" parameterization, and we want y to do what the negative of what y does in the parameterization. For now we'll guess that

x(t) = cos t
y(t) = -sin 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.