Need help with Newtonian equations with angular momentum

[Blockade]

For an equations such as this what goes into the θ?

θ = sinθ or θ = θ?

Let's say if the angle of displacement = 45° do I just plug 45° as θ into the equation below or should it be sin(45°)?

Or is it θ = S/R ?
ω_f² = ω_i² + 2 α (θ_f - θ_i)

 

[Simon Bridge]

There is no way of knowing without the specific problem you are trying to solve.

$\theta = \sin\theta$ is true for the intersection of the line $y=\theta$ with the curve $y=\sin\theta$ which occurs for $\theta=0$
The relation $\sin\theta \approx\theta$ is known as the par-axial approximation, it applies when $\theta \approx 0$.

If you want to find the arclength subtended by 45 degrees, then you would put $\theta = \frac{\pi}{4}$ into $S=R\theta$ to find out.

If an object has turned through 45deg starting with speed $\omega_i$ and accelerating at constant $\alpha$ ... and you wanted to know the final angular velocity, then you would put $\theta = \frac{\pi}{4}$ into $\omega_f^2=\omega_i^2+2\alpha\theta$

It is unclear what you mean by "the Newtonian equations" in this context ... Newton's second law, for instance, would be $\sum\vec \tau = I\vec\alpha$

In physics: angles are always used in radians, and you should never try to work a physics problem just by putting numbers into equations: you should use physics to find the equation first.