I was not having success at a trig solution.

I wrote a program that loops through 2 angles, calculating the third angle and finds the smallest error between the calculated function and the goal of 450/29

Results:

smallest error     0.00030537362685656433

numerator/29 (close to 450) 450.00885583517885

Angles A,B,C [43.6, 0.1, 136.3]

sin of angle A 0.6896195437356698

Putting that number into Wolfram Alpha finds:

sin((109 π)/450) ≈ 0.68961954373566978308

-----

import math

def sin(x):

    return math.sin(x*math.pi/180)

def cos(x):

    return math.cos(x*math.pi/180)

def f(A,B):

    """ return value of function a(cos A)+b(cos C)+c(cos B). Angles A and B are inputs, C is calculated.    A,B,C  are angles in degrees,  a,b,c are side lengths opposite the angles """

    C = 180 - A - B

    if A<=0 or B<=0 or C<=0 or A>=180 or B>=180 or C>=180:

        return 0

    a = 9

    b = a * sin(B)/sin(A)

    c = a * sin(C)/sin(A)

    return  a*cos(A) + b*cos(C) + c*cos(B)

goal = 450/29

leasterr = 10000

bestfunc = 10000

bestABC = []

for x in range(1,900): 

    for y in range(1,900):

        A = x/10

        B = y/10

        if A+B >=90:

            continue

        C = 180 - A - B

        if A<=0 or B<=0 or C<=0 or A>=180 or B>=180 or C>=180:

            continue

        func = f(A,B)

        err = abs(goal - func)

        if err < .0001:

            print(A,B,C,goal, sin(A))

        if err < leasterr:

            leasterr = err

            bestfunc = func

            bestABC = [A,B,C]

print('smallest error    ',leasterr)

print('numerator/29 (close to 450)', bestfunc*29)

print('Angles A,B,C',    bestABC)

print('sin of angle A',  sin(bestABC[0]) )