Would displacement vs time of circular motion be a $sin/cos$ curve or parabola?

I assuming that you wish to find the distance from the thrower as a function of time.

Suppose the thrower is at the origin, and the boomerang moves anti-clockwise in a circle of radius a with centre at (0,a) with angular speed $omega$. Then the position of the boomerang at time t is (x,y) where
$x = asin(omega t)$
$y = a(1-cos(omega t))$.
The distance from the origin is r where
$r^2 = x^2 + y^2$
$= a^2(sin^2(omega t)+1-2cos(omega t)+cos^2(omega t))$
$= 2a^2(1-cosomega t)$
$= (2asinfrac{omega t}{2})^2$
$r = |2asin(frac{1}{2} omega t)|$.
The graph of r vs t is a sine with amplitude $2a$ and frequency $frac{1}{2}omega$.