Explanation:
first, the law of cosine (the rule of Pythagoras generalized for any type of triangle) :
c² = a² + b² - 2ab×cos(C)
c is the side opposite of the angle C, a and b are the other 2 sides.
in our case :
b² = 5² + 8² - 2×5×8×cos(51)
b² = 25 + 64 - 80×cos(51) =
= 89 - 80×cos(51) = 38.65436872...
b = 6.217263764... ≈ 6.22
now we have all 3 sides and need to find the other 2 angles.
law of sine
a/sin(A) = b/sin(B) = c/sin(C)
a, b, c are the sides, and A, B, C are the corresponding opposite angles.
5/sin(A) = 6.217263764.../sin(51)
sin(A) = 5×sin(51)/6.217263764... =
= 0.624990342...
A = 38.6814786...° ≈ 38.68°
sin(C) = 8×sin(51)/6.217263764... =
= 0.999984547...
C = 89.6814786...° ≈ 89.68°