How to get the first term of a Taylor Series? [duplicate]

Question

How can I get the first non-zero term of a Taylor Series, when I don't know beforehand what power it will be? Example:

Series[Sin[a]^6],{a,0,6}]

returns $a^6$, but I have to know the order is 6. If I instead just say:

Series[Sin[a]^6],{a,0,1}]

I get back $O[a]^6$ but I don't know what command to use to extract the 6 from that to redo the Seres[] calculation.

The function I want would just work like this:

f[(Sin[a*Cos[a]]^7)*Cos[2a]^3]

$a^7$

Any ideas?

Answer 1

Asymptotic (available from 12.1, I think) works for your first example:

Asymptotic[Sin[a]^6, a -> 0]
(* a^6 *)