


from __future__ import annotations

from numpy import array, cos, cross, floataa, radians, sin
from numpy.typing import NDArray




def  lowerCAmelCase_ (							UpperCamelCase_					,		UpperCamelCase_					,		UpperCamelCase_ = False  )     ->		list[float]:
       if radian_mode:
              return [magnitude * cos(UpperCamelCase_  ), magnitude * sin(UpperCamelCase_  )]
       return [magnitude * cos(radians(UpperCamelCase_  )  ), magnitude * sin(radians(UpperCamelCase_  )  )]




def  lowerCAmelCase_ (							UpperCamelCase_					,		UpperCamelCase_					,		UpperCamelCase_ = 10**-1  )     ->		bool:
       UpperCamelCase_          =  cross(UpperCamelCase_					,		UpperCamelCase_  )
       UpperCamelCase_          =  sum(UpperCamelCase_  )
       return abs(UpperCamelCase_  ) < eps


if __name__ == "__main__":
     # Test to check if it works
     _UpperCAmelCase        =				array(
         [
             polar_force(718.4, 1_8_0 - 3_0),
             polar_force(879.54, 4_5),
             polar_force(1_0_0, -9_0),
         ]
     )

     _UpperCAmelCase        =				array([[0, 0], [0, 0], [0, 0]])

     assert in_static_equilibrium(forces, location)

     # Problem 1 in image_data/2D_problems.jpg
     _UpperCAmelCase        =				array(
         [
             polar_force(3_0 * 9.81, 1_5),
             polar_force(2_1_5, 1_8_0 - 4_5),
             polar_force(2_6_4, 9_0 - 3_0),
         ]
     )

     _UpperCAmelCase        =				array([[0, 0], [0, 0], [0, 0]])

     assert in_static_equilibrium(forces, location)

     # Problem in image_data/2D_problems_1.jpg
     _UpperCAmelCase        =				array([[0, -2_0_0_0], [0, -1_2_0_0], [0, 1_5_6_0_0], [0, -1_2_4_0_0]])

     _UpperCAmelCase        =				array([[0, 0], [6, 0], [1_0, 0], [1_2, 0]])

     assert in_static_equilibrium(forces, location)

     import doctest

     doctest.testmod()








import sys




def  lowerCAmelCase_ (							UpperCamelCase_  )     ->		Union[str, Any]:
       UpperCamelCase_          =  len(UpperCamelCase_  )
       UpperCamelCase_          =  [[0 for x in range(UpperCamelCase_  )] for x in range(UpperCamelCase_  )]
       UpperCamelCase_          =  [[0 for x in range(UpperCamelCase_  )] for x in range(UpperCamelCase_  )]

       for chain_length in range(2					,		UpperCamelCase_  ):
              for a in range(1					,		n - chain_length + 1  ):
                     UpperCamelCase_          =  a + chain_length - 1

                     UpperCamelCase_          =  sys.maxsize
                     for c in range(UpperCamelCase_					,		UpperCamelCase_  ):
                            UpperCamelCase_          =  (
                                matrix[a][c] + matrix[c + 1][b] + array[a - 1] * array[c] * array[b]
                            )
                            if cost < matrix[a][b]:
                                   UpperCamelCase_          =  cost
                                   UpperCamelCase_          =  c
       return matrix, sol




def  lowerCAmelCase_ (							UpperCamelCase_					,		UpperCamelCase_					,		UpperCamelCase_  )     ->		List[str]:
       if i == j:
              print("A" + str(UpperCamelCase_  )					,		end=" "  )
       else:
              print("("					,		end=" "  )
              print_optiomal_solution(UpperCamelCase_					,		UpperCamelCase_					,		optimal_solution[i][j]  )
              print_optiomal_solution(UpperCamelCase_					,		optimal_solution[i][j] + 1					,		UpperCamelCase_  )
              print(")"					,		end=" "  )




def  lowerCAmelCase_ (							)     ->		Any:
       UpperCamelCase_          =  [30, 35, 15, 5, 10, 20, 25]
       UpperCamelCase_          =  len(UpperCamelCase_  )
       # Size of matrix created from above array will be
       # 30*35 35*15 15*5 5*10 10*20 20*25
       UpperCamelCase_       , UpperCamelCase_          =  matrix_chain_order(UpperCamelCase_  )

       print("No. of Operation required: " + str(matrix[1][n - 1]  )  )
       print_optiomal_solution(UpperCamelCase_					,		1					,		n - 1  )


if __name__ == "__main__":
     main()

