How do you eliminate the parameter theta to find a Cartesian equation of the curve: x=sin (1/2 theta), y=cos (1/2 theta), 0 is less than or equal to theta and theta is less than or equal to 4pi

The answer to the problem is as follows:

x = sin (t/2)

y = cos (t/2)

Square both equations and add to eliminate the parameter t:

x^2 + y^2 = sin^2 (t/2) + cos^2 (t/2) = 1

The final step is translating the original parameter limits into limits on x and y. Over the - Pi to + Pi range of t, x varies from - 1 to + 1, whereas y varies from 0 to 1. Thus we have the semicircle in quadrants I and II: y > = 0.