Deriving height(time) for a ferris wheel passenger? I'm trying to figure this out algebraically deriving $h(t)$ 
I know it's a trig function and by trig: $opposite = adjacent * tan(θ)$ 
The length of the adjacent side and value of θ are related to a component of the velocity which is related to the circumference of the circle I imagine: $v ∝ \frac{2πr}{t} $
This is about all I have so far, and I'm trying to work it out. Not for homework, my homework was about Ferris Wheels but as it turns out figuring this out is more challenging and fun, just I'm stumped still.
Any tips/help?
 A: If you are just looking for the height as a function of time, you indeed want a trigonometric function but not $\tan$ because it varies from $-\infty$ to $\infty$. Instead, you want another trigonometric function like $\sin$ or $\cos$ that varies from $-1$ to $1$ which can be mapped to the bottom and top of the ferris wheel.
Let $t = 0$ be the initial time at the bottom of the ferris wheel at height $0$ and $t = 1$ be the time for half a revolution at the top of the ferris wheel at height $2r$ where $r$ is the radius of the ferris wheel. Given $\cos(0) = 1$ and $\cos(\pi) = -1$, these values are mapped to $0$ at $t = 0$ and $2r$ at $t = 1$ respectively:
$$
h(t) = 2r(1 - cos(t\pi)) \\
h(0) = 0 \\
h(1) = 2r \\
$$
Similarly, $\sin$ could be mapped by applying the complement rule:
$$
h(t) = 2r(1 - sin(\pi/2 - t\pi)) \\
h(t) = 2r(1 - sin(\pi/2(1 - 2t))) \\
$$
If you are also looking for the speed at which the outer circumference is moving, you don't really need a trigonometric function. Instead, you can consider the definition of a radian as the length of a corresponding arc of a unit circle. Given $t$ as the unit for half a revolution and $2\pi r$ as the radians for a complete revolution, the distance traveled by the circumference of the ferris wheel as a function of $t$ can be expressed as:
$$
p(t) = t \pi r \\
$$
The speed of the outer circumference can be measured by differentiating $p(t)$ to $p'(t) = \pi r$. Note that the unit is not necessarily per seconds as $t$ was defined arbitrarily based on half a revolution. See the comments for details to change the unit.
