diff --git a/sympy/simplify/fu.py b/sympy/simplify/fu.py
index 8cd518b3d4e1..bd66c2c16193 100644
--- a/sympy/simplify/fu.py
+++ b/sympy/simplify/fu.py
@@ -246,6 +246,13 @@ def f(rv):
             rv = fmap[type(rv)](S.Pi/2 - rv.args[0])
         return rv
 
+    # touch numbers iside of trig functions to let them automatically update
+    rv = rv.replace(
+        lambda x: isinstance(x, TrigonometricFunction),
+        lambda x: x.replace(
+            lambda n: n.is_number and n.is_Mul,
+            lambda n: n.func(*n.args)))
+
     return bottom_up(rv, f)
 
 
@@ -273,7 +280,12 @@ def TR4(rv):
     0 1 zoo 0
     """
     # special values at 0, pi/6, pi/4, pi/3, pi/2 already handled
-    return rv
+    return rv.replace(
+        lambda x:
+            isinstance(x, TrigonometricFunction) and
+            (r:=x.args[0]/pi).is_Rational and r.q in (1, 2, 3, 4, 6),
+        lambda x:
+            x.func(x.args[0].func(*x.args[0].args)))
 
 
 def _TR56(rv, f, g, h, max, pow):
diff --git a/sympy/simplify/tests/test_fu.py b/sympy/simplify/tests/test_fu.py
index 29f614d7ad6f..0b0a6fbc491b 100644
--- a/sympy/simplify/tests/test_fu.py
+++ b/sympy/simplify/tests/test_fu.py
@@ -1,15 +1,17 @@
 from sympy.core.add import Add
 from sympy.core.mul import Mul
 from sympy.core.numbers import (I, Rational, pi)
+from sympy.core.parameters import evaluate
 from sympy.core.singleton import S
 from sympy.core.symbol import (Dummy, Symbol, symbols)
 from sympy.functions.elementary.hyperbolic import (cosh, coth, csch, sech, sinh, tanh)
 from sympy.functions.elementary.miscellaneous import (root, sqrt)
-from sympy.functions.elementary.trigonometric import (cos, cot, csc, sec, sin, tan)
+from sympy.functions.elementary.trigonometric import (cos, cot, csc, sec, sin, tan,
+    TrigonometricFunction)
 from sympy.simplify.powsimp import powsimp
 from sympy.simplify.fu import (
     L, TR1, TR10, TR10i, TR11, _TR11, TR12, TR12i, TR13, TR14, TR15, TR16,
-    TR111, TR2, TR2i, TR3, TR5, TR6, TR7, TR8, TR9, TRmorrie, _TR56 as T,
+    TR111, TR2, TR2i, TR3, TR4, TR5, TR6, TR7, TR8, TR9, TRmorrie, _TR56 as T,
     TRpower, hyper_as_trig, fu, process_common_addends, trig_split,
     as_f_sign_1)
 from sympy.core.random import verify_numerically
@@ -72,6 +74,17 @@ def test_TR3():
         j = TR3(i)
         assert verify_numerically(i, j) and i.func != j.func
 
+    with evaluate(False):
+        eq = cos(9*pi/22)
+    assert eq.has(9*pi) and TR3(eq) == sin(pi/11)
+
+
+def test_TR4():
+    for i in [0, pi/6, pi/4, pi/3, pi/2]:
+        with evaluate(False):
+            eq = cos(i)
+        assert isinstance(eq, cos) and not isinstance(TR4(eq), cos)
+
 
 def test__TR56():
     h = lambda x: 1 - x