Theta is Also a Sinusoid

Let $v_a$ be the average speed and $s$ be the travel space and the time is $t = \frac{\pi}{2} - \frac{-\pi}{2} = \pi$ . Then, $v_a = \frac{\text{travel space}}{time} = \frac{s}{t}$ . Since $\sin(t) \in [- 1, 1], \space \text{ cos(sin(t))}$ always will be positive, $v_a = \frac{s}{t} = \frac{\int ds}{t} = \frac{\int ((dx)^2 + (dy)^2)^{\frac{1}{2}}}{t} =$ $= \int_{-\pi/2}^{\pi/2} \frac{\left(\cos^2(t) \cdot (\sin^2(\sin(t)) + \cos^2(\sin(t))\right)^{\frac{1}{2}}}{\pi} dt = \int_{-\pi/2}^{\pi/2} \frac{|\cos(t)|}{\pi} dt =$ $= (\frac{1}{\pi})\cdot \left(\sin(t) \right)_{-\pi/2}^{\pi/2} = \frac{2}{\pi}$

One can certainly use calculus to solve, but it's also apparent that in the interval $t = -\frac{\pi}{2}$ to $t = \frac{\pi}{2}$ , $sin(t)$ goes from -1 radian to 1 radian. Therefore, we have 2 radians per $\pi$ units of time.