Why is running this code always displaying Running?

Question

Clear["Global`*"]
Maximize[{Tan[a], a + b + c == \[Pi], 
   Sin[a] + 3 Sin[b] Cos[c] == 0, \[Pi]/2 < c < \[Pi], 
   0 < a < \[Pi]/2, 0 < b < \[Pi]/2}, {a, b, c}] // FullSimplify

Why is running this code always displaying Running? No calculation results

Answer 1

We calculate the max value of a, then Tan[a] since Tan is increasing in 0 < a < π/2

max=Maximize[{a, a + b + c == π, 
  Sin[a] + 3 Sin[b] Cos[c] == 0, π/2 < c < π, 0 < a < π/2,
   0 < b < π/2}, {a, b, c}]
Tan[max // First] // FullSimplify

3/4

Answer 2

Or use a numeric method

$Version

(* "13.2.1 for Mac OS X ARM (64-bit) (January 27, 2023)" *)

Clear["Global`*"]

{max, arg} = 
 NMaximize[{Tan[a], a + b + c == π, 
     Sin[a] + 3 Sin[b] Cos[c] == 0, π/2 < c < π, 
     0 < a < π/2, 0 < b < π/2}, {a, b, c}, 
    WorkingPrecision -> 30] // N // Rationalize

(* {3/4, {a -> 0.643501, b -> 0.463648, c -> 2.03444}} *)