Give parametric equations and bounds for the parameter that describe the unit circle as shown. In each case the unit circle should be traced only once. Check the answers by putting them in a calculator and seeing if we find the right picture.

Answer

- We want to start at (0,1) and go around the circle counterclockwise from t = 0 to t = 4π.

From this example we know that switching the "normal" equations traces out the circle clockwise startingfrom (0,1).

*x*(*t*)& = &sin t*y*(*t*)& = &cos t

0≤ t ≤ 2πPICTURE param eq 23Start with these equations. We want *y* to stay the same, moving from 1 down to $-1 and back up to $1:PICTURE param eq 24However, we want *x* to be the opposite. We want *x* to first move to $-1, then to $ + 1, then back to 0:PICTURE param eq 25If we stick a negative sign in front of the *x* equation and leave the *y* equation unchanged, we'll draw the circle in the correct direction. The equations

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

for 0 ≤ t ≤ 2πproduce the circle starting at the correct point and drawn in the correct direction:PICTURE graph these, with arrows and label t = 0 and t = 2πThere's one thing left to address: the speed. Right now we're taking from 0 to $2π$ to draw the circle. That's not long enough. We want to take from 0 to 4π. Tofix this, double the period of each equation. The final parameterization is

*x*(*t*) = -sin (\frac{t}{2})*y*(*t*) = cos (\frac{t}{2})

for 0 ≤ t ≤ 4π.