I trying to see some tricks to simplify some functions, for instance:
$$\sin(\alpha x)\approx\alpha x$$ for small values of $x$.
I'm wondering if there is a table with these approximations, or a rigorous way to get them as I have noticed that they are essential to easily solve some exercises.
Moreover, I am also convinced that they derive from Taylor's series of these functions, but I have not been able to get them right from the beginning.
Thank you very much.
The Taylor series and its approximations for small values of $x$ would be:
$$\sin(x)= x - \frac {x^3}6 + \frac {x^5}{120} - ... \approx x $$
$$\cos(x)= 1 - \frac {x^2}2 + \frac {x^4}{24} - ... \approx 1 $$
$$\tan(x) \approx \sin(x) \approx x$$
It is readily seen that for the sine function the second most significant (third-order) term falls off as the cube of the first term; thus, even for a not-so-small argument such as $0.01$, the value of the second most significant term is on the order of $0.000001$, or $\frac 1{10000}$ the first term.
So every time when you start to integrate or take a derivative and let $\Delta x \to 0$, the error is negligible.