Finding a parametric Equation for a Cycloid

**lilaziz1** · April 8th 2010, 09:55 AM

"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

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

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

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

Thus,

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

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

**dwsmith** · April 8th 2010, 05:31 PM

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

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

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

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