functions - Why do certain values in ArcSin not fully evaluate?

This is a continuation of: Why do certain fractional values in TriangleWave not evaluate?
The analysis by R.M and rasher revealed that problem to reduce to the behavior of:

ArcSin[Sin[π / 10]]

ArcSin[1/4 (-1 + Sqrt[5])]

Every other denominator besides ten, at least up to 500,000, automatically evaluates to a simpler form:

 Position[ArcSin[Sin[π / Range[500000]]], _ArcSin]  (* slow *)

{{10}}

All other coefficient of π with a denominator of ten appear to exhibit the same problem:

ArcSin[Sin[{1, 3, 7, 9, 11} π/10]]

{ArcSin[1/4 (-1 + Sqrt[5])], ArcSin[1/4 (1 + Sqrt[5])], ArcSin[1/4 (1 + Sqrt[5])], 
 ArcSin[1/4 (-1 + Sqrt[5])], ArcSin[1/4 (1 - Sqrt[5])]}

FullSimplify reduces the expression:

 ArcSin[Sin[{1, 3, 7, 9, 11} π/10]] // FullSimplify

{π/10, (3 π)/10, (3 π)/10, π/10, -(π/10)}

Is this a bug? Is there some explanation for it?

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Before someone points out the obvious: I see that Sin[π/10] evaluates to 1/4 (-1 + Sqrt[5]). But why isn't ArcSin "smart" enough to recognize this value, when it recognizes others? For example Sin[π/12] evaluates to (-1 + Sqrt[3])/(2 Sqrt[2]) but ArcSin correctly recognizes this and outputs π/12.