# Thread: Finding a parametric Equation for a Cycloid

1. ## Finding a parametric Equation for a Cycloid

"We suppose that the wheel rolls to the right. P being at the origin when the turn angle t equals 0. The figure shows the wheel after it has turned t radians. The base of the wheel is at distance at from the origin. The wheel's center is at (at,a), and the coordinates of P are

$\displaystyle x = at + a\cos\theta, y = a + a\sin\theta.$

To express $\displaystyle \theta$ in terms of t, we absorve that $\displaystyle t + \theta = 3\pi/2 + 2k\pi$ for some integer k, so

$\displaystyle \theta = 3\pi/2 - t + 2k\pi$

Thus,

$\displaystyle \cos\theta = \cos(\frac{3\pi}{2} - t + 2k\pi) = -\sin t$

I get everything up to the prior line. How does that equal -sint?

2. $\displaystyle cos(x-\frac{\pi }{2})=sin(x)$

Your equation is of the form $\displaystyle cos(\frac{\pi }{2}-x)=-sin(x)$.

Assuming k is an integer. This $\displaystyle \frac{3\pi }{2}+2k\pi$ always yields a $\displaystyle \frac{-\pi }{2}$