If theta is considered small in problems

[M. next]

we know when theta is considered small enough, sin(theta) is approximately equal to theta. Ok, the question is what happens to theta dot(the time derivative of theta)?

 

[mfb]

If $\theta$ is small all the time, the time-derivative of $sin(\theta)$ is approximately the time-derivative of $\theta$.

 

[M. next]

thanks

 

[Nabeshin]

Another way of seeing the above is that the time derivative of sin is cosine, and when theta is small cosine is approximately equal to one.

 

[pmacias]

As theta is considered small enough, the approximation of sin(theta) being equal to theta becomes more accurate. Therefore, the time derivative of theta, theta dot, will also approach the same value as sin(theta) and become approximately equal to theta dot. This is because the time derivative measures the rate of change of a variable, and as theta becomes smaller and closer to its true value, the rate of change also becomes smaller and approaches the true value of theta dot.