In Mathematica I have In[181]:= FullSimplify[ArcSin[10^(1/2)] == (Pi/2 - ArcSinh[3] I)] Out[181]= True
In[206]:= FullSimplify[ArcSin[100^(1/3)] == (Pi/2 - ArcCosh[10^(2/3)] I)] Out[206]= True
In[198]:= FullSimplify[ArcSin[1000^(1/4)] == (Pi/2 - ArcSinh[Sqrt[10 Sqrt[10] - 1]] I)] Out[198]= True
In[205]:= FullSimplify[ArcSin[10000^(1/5)] == (Pi/2 - ArcCosh[10^(4/5)] I)] Out[205]= True
If I translated what I have correctly it means that the following table of exact values is true, with emphasis on the imaginary parts.
$$ \begin{array}{| c | r |} \hline n& arcsin(n) \\ \hline \\ \hline 10^{1/2}&Pi/2-arcsinh(10*.3)I \\ \hline 100^{1/3}& Pi/2-arccosh(10^{2/3})I \\ \hline 1000^{1/4}& Pi/2-arcsinh(\sqrt{10 \sqrt{10} - 1})I \\ \hline 10000^{1/5}& Pi/2-arccosh(10^{4/5})I \\ \hline 100000^{1/6}&????????????????????? \\ \hline \end{array} $$ I expected some such exact values for the roots of the powers of 10 from the following question. Help explain a new theory on small sines
But why these values?