How sinusoidal wave relates with wheel mathematically?

Question

Wheel has parameters $v$ velocity, $r$ radius, and the sinusoidal has $a$ amplitude and $f$ is frequency.

Answer 1

Suppose we have a wheel with radius $R$ spinning with angular frequency $\omega$ in time. Then, a point $P$ on the rim of the wheel forms an angle $\theta(t)$ with the positive $x$-axis such that $\theta(t) = \omega t$. Hence, this point can be described with the parametric equations $$ x(t) = R\cos\omega t \\ y(t) = R\sin\omega t. $$ If we plot either of $x(t)$ or $y(t)$ as a function of time, the resulting graph is a sinusoidal wave, as we can clearly see based on these parametric equations, while the plot of $P(t)$ would be a circle of radius $R$.

The velocity of the point $P(t)$ is $\dot P(t) = \omega R(-\sin\omega t,\cos\omega t)$, and the linear speed of the point $P(t)$ is $$ v = \sqrt{(-\omega R \sin\omega t)^2 + (\omega R\cos\omega t)^2} = \omega R $$ where we have made use of the identity $\sin^2x + \cos^2x = 1$.

Answer 2

Assume a point on the unit circle. In the following picture you can see the relation of the angle that the pint makes with origin $(\theta)$ and the two functions $\sin(\theta)$ and $\cos(\theta)$

You can use this to understand the answer by AOrtiz.