1. ## Please discuss: Relating angular velocity to a sinusoidal function

I recently did an angular velocity problem where they gave the angular velocity (rate of rotation) and asked me to write it as a cosine function of the form:

f(x) = a * cos [k (x - d)] + c

A tutor told me that "k", the constant of horizontal compression, is also the angular velocity, so that:

Period = (2pi) / abs(k)

This means that the sinusoidal function is related to a circle and can be used to model angular velocity.

Is there some theorem that relates the sinusoidal function to the unit circle? I am interested in knowing about this, and google returned nothing relevant. I would appreciate some further research into this.

2. ## Re: Please discuss: Relating angular velocity to a sinusoidal function

using $x$ and $y$ positions as functions of time for an object moving in a circle of radius $r$ centered at the origin, the following set of parametric equations model the uniform circular motion in two dimensions ...

$x = r\cos(\omega t)$

$y = r\sin(\omega t)$

... where the starting position is $(r,0)$ at time $t=0$ and angular velocity, $\omega = \dfrac{2\pi}{T}$

animation link where one may adjust $r$ and $T$

Uniform circular motion