"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?