Question

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

Answer 1

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.