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.
We want to start at (-1,0) and go around the circle clockwise. The value of y will be doing the same thing it always does: going from 0 up to 1, down to -1, then back up to 0.
We can leave the equation for y unchanged.
The value of x usually goes from 1 to -1 and back to 1.
Instead, we want to go from -1 to 1 and back to -1.
That is, we want x to be the negative of what it usually is. We need to stick a negative sign on the front of the equation for x.
The new parameterization isx(t) = -cos ty(t) = sin t.Since we're going around the circle at the normal speed, we have0≤ t ≤ 2π.