See the tension in the string with the mgsinθ from the horizontal
At the vertical T = mv2/r + mgsinθ
Max T =2mg
Now we consider the pendulum is left from the horizontal position therefore v = √2gh
sin θ = h / L
h = Lsinθ
v = √2gLsinθ
Therefore
2mg = m * 2gLsinθ/L + mgsinθ
2 = 2sinθ + sinθ
2 = 2sinθ + Sinθ
Sinθ = 2/3