Trivial simplifications for arccos

[rhinopotamus]

This fixes a couple of places where arccos doesn't automatically simplify for ratios arising from the unit circle (but arcsin does).

Turns out that in src/sage/symbolic/ginac/inifcns_trig.cpp, asin_eval has some logic handling unit circle cases directly that was not replicated in acos_eval. I copy-pasted the logic from asin_eval and modified appropriately.

Partially fixes #24211.

I'm not sure exactly how to create tests covering this change but would be happy to do so with a little guidance!

📝 Checklist

• The title is concise, informative, and self-explanatory.
• The description explains in detail what this PR is about.
• I have linked a relevant issue or discussion.
• I have created tests covering the changes.
• I have updated the documentation accordingly.

⌛ Dependencies

None!

[github-actions]

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[tscrim]

You should be able to add doctests showing the simplifications are occurring in the output. I imagine

sage: arcsin(sqrt(2)/2)
1/4*pi
sage: arccos(sqrt(2)/2)  # is simplified to 1/4*pi with your change
arccos(1/2*sqrt(2))

[fchapoton]

one test has to be fixed:

File "src/doc/de/thematische_anleitungen/sage_gymnasium.rst", line 790, in doc.de.thematische_anleitungen.sage_gymnasium
Failed example:
    acos(sqrt(3)/2)
Expected:
    arccos(1/2*sqrt(3))
Got:
    1/6*pi

and the same in src/sage/tests/books/computational-mathematics-with-sagemath/premierspas_doctest.py

[tscrim]

LGTM. Just will wait for a green bot before a positive review.

[tscrim]

Trivial failures in

src/sage/geometry/hyperbolic_space/hyperbolic_geodesic.py
src/sage/tests/books/computational-mathematics-with-sagemath/premierspas_doctest.py

[fchapoton]

the failures are not so trivial to fix, and keep their meaning

[fchapoton]

From ad4c750bfb1ab05a2bcf06247e1baec44506fb58 Mon Sep 17 00:00:00 2001
From: =?UTF-8?q?Fr=C3=A9d=C3=A9ric=20Chapoton?= <[email protected]>
Date: Tue, 4 Feb 2025 21:51:54 +0100
Subject: [PATCH] fixing the doctests

---
 src/doc/de/thematische_anleitungen/sage_gymnasium.rst     | 8 +-------
 src/sage/geometry/hyperbolic_space/hyperbolic_geodesic.py | 4 ++--
 .../premierspas_doctest.py                                | 2 +-
 3 files changed, 4 insertions(+), 10 deletions(-)

diff --git a/src/doc/de/thematische_anleitungen/sage_gymnasium.rst b/src/doc/de/thematische_anleitungen/sage_gymnasium.rst
index 9dc38d8bf0b..dfeb3e13c82 100644
--- a/src/doc/de/thematische_anleitungen/sage_gymnasium.rst
+++ b/src/doc/de/thematische_anleitungen/sage_gymnasium.rst
@@ -774,8 +774,7 @@ falls wir im Bogenmass rechnen möchten. Ansonsten müssen wir wie oben beschrie
 
 Ihre Umkehrfunktionen sind auch mit den nicht sehr überraschenden Namen ``asin()``, ``acos()``, ``atan()`` und ``acot()`` versehen.
 Sie geben uns aber wie oben erklärt nur Winkel im Bogenmass zurück. Möchten wir im Gradmass rechnen, müssen wir wieder
-konvertieren. Die exakte Berechnung der Werte funktioniert in die Gegenrichtung nur, falls im ursprünglichen Wert keine
-Wurzeln vorkommen::
+konvertieren. Exakte Berechnung der Werte funktioniert, wenn es möglich ist::
 
     sage: atan(1)
     1/4*pi
@@ -784,12 +783,7 @@ Wurzeln vorkommen::
     sage: rad2deg(x) = x*(180/pi)
     sage: rad2deg(acos(-1/2))
     120
-
-Falls wir Wurzelterme verwenden, müssen wir mit der Funktion ``simplify_full()`` vereinfachen::
-
     sage: acos(sqrt(3)/2)
-    arccos(1/2*sqrt(3))
-    sage: (acos(sqrt(3)/2)).simplify_full()
     1/6*pi
 
 Sage kann auch weitere Regeln für trigonometrische Funktionen anwenden, um Terme zu vereinfachen. Es kennt zum Beispiel auch die
diff --git a/src/sage/geometry/hyperbolic_space/hyperbolic_geodesic.py b/src/sage/geometry/hyperbolic_space/hyperbolic_geodesic.py
index b9488e77b16..52f889f30d1 100644
--- a/src/sage/geometry/hyperbolic_space/hyperbolic_geodesic.py
+++ b/src/sage/geometry/hyperbolic_space/hyperbolic_geodesic.py
@@ -1854,9 +1854,9 @@ class HyperbolicGeodesicUHP(HyperbolicGeodesic):
             sage: g = HyperbolicPlane().UHP().get_geodesic(1, 1 + I)
             sage: h = HyperbolicPlane().UHP().get_geodesic(-sqrt(2), sqrt(2))
             sage: g.angle(h)
-            arccos(1/2*sqrt(2))
+            1/4*pi
             sage: h.angle(g)
-            arccos(1/2*sqrt(2))
+            1/4*pi
 
         Angle is unoriented, as opposed to oriented. ::
 
diff --git a/src/sage/tests/books/computational-mathematics-with-sagemath/premierspas_doctest.py b/src/sage/tests/books/computational-mathematics-with-sagemath/premierspas_doctest.py
index feb5391b4d6..027c31eecfe 100644
--- a/src/sage/tests/books/computational-mathematics-with-sagemath/premierspas_doctest.py
+++ b/src/sage/tests/books/computational-mathematics-with-sagemath/premierspas_doctest.py
@@ -65,7 +65,7 @@ Sage example in ./premierspas.tex, line 952::
 Sage example in ./premierspas.tex, line 967::
 
   sage: arccos(sin(pi/3))
-  arccos(1/2*sqrt(3))
+  1/6*pi
   sage: sqrt(2)
   sqrt(2)
   sage: exp(I*pi/7)
-- 
2.43.0