Asymptotic approximation/expansion for arccosine function?

Question

Trying to find a 3-term asymptotic expansion for $z=\cos^{-1}(x)$, as $x\to 1^-$. Found a lot of examples online for inverse tangent, cosine, etc. but have yet to find any guidance on the inverse cosine function. Any help would be greatly appreciated.

Answer 1

This is equivalent to the expansion of $\arccos(1-t)$ at $t=0^{+}$.

Ask Wolfram Alpha Series[Arccos[1-t)],{t,0,3}]and re-substitute $t=1-x$ in the result $$\arccos(1-t) =\sqrt{2} \sqrt{t} + \frac{t^{3/2}}{6 \sqrt{2}} + \frac{3t^{5/2}}{80 \sqrt{2}} + \frac{5 t^{7/2}}{448 \sqrt{2}} + O(t^{9/2})$$ or see 1.

Yet another approach would be do use the trigonometric identity from 2 and get $$\arccos(1-t) = 2\arcsin\left(\sqrt{t/2}\right),$$ Then use the expansion of $\arcsin.$