Derivation of angular velocity using the unit circle

[jnuz73hbn]

Homework Statement

I am looking for a good derivation of $$ \ ω = \frac{Δα}{Δt} \ $$
, starting from the unit circle. My approach would be to first construct a right-angled triangle (Pythagorean theorem), then express $$ cos(α) $$ for the ankathete and $$ sin(α) $$ as the anticathete. Then I have a point on the arc of the circle (r=1). How do I get a suitable derivation for the initial formula of the angular velocity?

Relevant Equations

$$ \ Δα = α_2 - α_1 \ $$

$$ \ ω = \frac{Δα}{Δt} \ $$

 

[PeroK]

Homework Statement: I am looking for a good derivation of $$ \ ω = \frac{Δα}{Δt} \ $$

That's essentiallly the definition, although usually $\theta$ or $\phi$ is used as the polar angle. The definition of angular velocity is $$\omega = \lim_{\Delta t \to 0} \frac{\Delta \theta}{\Delta t} = \frac{d\theta}{dt}$$See, for example:

https://en.wikipedia.org/wiki/Angular_velocity

 

[jnuz73hbn]

That's essentiallly the definition, although usually $\theta$ or $\phi$ is used as the polar angle. The definition of angular velocity is $$\lim_{\Delta t \to 0} \frac{\Delta \theta}{\Delta t}$$See, for example:

https://en.wikipedia.org/wiki/Angular_velocity

however, i wanted to go via the unit circle with sinus and cosine to derive exactly this definition

 

[PeroK]

however, i wanted to go via the unit circle with sinus and cosine to derive exactly this definition

By definition, you can't derive a definition.

 

[jnuz73hbn]

By definition, you can't derive a definition.

I just want to know where the formula comes from using the unit circle or how it relates to sin cos in the unit circle

 

[PeroK]

I just want to know where the formula comes from using the unit circle or how it relates to sin cos in the unit circle

In plane polar coordinates, angular velocity $\omega$ is defined as $\omega = \frac{d\theta}{dt}$. This means that, for example, uniform circular motion about the origin is given by:
$$x = R\cos(\omega t), \ y = R\sin(\omega t)$$