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 ty(t) = -sin t
for 0 ≤ t≤ 2π.