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?

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.

Comments

functions - Get leading series expansion term?

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mathematical optimization - Minimizing using indices, error: Part::pkspec1: The expression cannot be used as a part specification

I want to use Minimize where the variables to minimize are indices pointing into an array. Here a MWE that hopefully shows what my problem is. vars = u@# & /@ Range[3]; cons = Flatten@ { Table[(u[j] != #) & /@ vars[[j + 1 ;; -1]], {j, 1, 3 - 1}], 1 vec1 = {1, 2, 3}; vec2 = {1, 2, 3}; Minimize[{Total@((vec1[[#]] - vec2[[u[#]]])^2 & /@ Range[1, 3]), cons}, vars, Integers] The error I get: Part::pkspec1: The expression u[1] cannot be used as a part specification. >> Answer Ok, it seems that one can get around Mathematica trying to evaluate vec2[[u[1]]] too early by using the function Indexed[vec2,u[1]] . The working MWE would then look like the following: vars = u@# & /@ Range[3]; cons = Flatten@{ Table[(u[j] != #) & /@ vars[[j + 1 ;; -1]], {j, 1, 3 - 1}], 1 vec1 = {1, 2, 3}; vec2 = {1, 2, 3}; NMinimize[ {Total@((vec1[[#]] - Indexed[vec2, u[#]])^2 & /@ R...

plotting - Plot 4D data with color as 4th dimension

I have a list of 4D data (x position, y position, amplitude, wavelength). I want to plot x, y, and amplitude on a 3D plot and have the color of the points correspond to the wavelength. I have seen many examples using functions to define color but my wavelength cannot be expressed by an analytic function. Is there a simple way to do this? Answer Here a another possible way to visualize 4D data: data = Flatten[Table[{x, y, x^2 + y^2, Sin[x - y]}, {x, -Pi, Pi,Pi/10}, {y,-Pi,Pi, Pi/10}], 1]; You can use the function Point along with VertexColors . Now the points are places using the first three elements and the color is determined by the fourth. In this case I used Hue, but you can use whatever you prefer. Graphics3D[ Point[data[[All, 1 ;; 3]], VertexColors -> Hue /@ data[[All, 4]]], Axes -> True, BoxRatios -> {1, 1, 1/GoldenRatio}]

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