To solve the triangle ABC with side lengths a = 54, b = 12, and c = 51, we can use the Law of Cosines and the Law of Sines.
1. Find angle A using the Law of Cosines:
cos(A) = (b^2 + c^2 - a^2) / (2 * b * c)
cos(A) = (12^2 + 51^2 - 54^2) / (2 * 12 * 51)
cos(A) = (144 + 2601 - 2916) / 1224
cos(A) = -0.0123
Since the cosine value is negative, angle A is an obtuse angle.
2. Find angle B using the Law of Cosines:
cos(B) = (a^2 + c^2 - b^2) / (2 * a * c)
cos(B) = (54^2 + 51^2 - 12^2) / (2 * 54 * 51)
cos(B) = (2916 + 2601 - 144) / 5544
cos(B) = 0.9817
3. Find angle C using the Law of Cosines:
cos(C) = (a^2 + b^2 - c^2) / (2 * a * b)
cos(C) = (54^2 + 12^2 - 51^2) / (2 * 54 * 12)
cos(C) = (2916 + 144 - 2601) / 1296
cos(C) = 0.0289
4. Find angle C using the Law of Sines:
sin(C) / c = sin(A) / a
sin(C) = (sin(A) / a) * c
sin(C) = (sin(A) / 54) * 51
sin(C) = (sqrt(1 - cos(A)^2) / 54) * 51
sin(C) = (sqrt(1 - (-0.0123)^2) / 54) * 51
sin(C) = 0.8682
Now, we have the values of cos(C) and sin(C). We can use these values to find angle C.
C = arcsin(sin(C))
C = arcsin(0.8682)
C = 60.7 degrees
5. Find angle A using the sum of angles in a triangle:
A = 180 - B - C
A = 180 - 77.5 - 60.7
A = 41.8 degrees
Now, we have found the three angles of the triangle:
A = 41.8 degrees
B = 77.5 degrees
C = 60.7 degrees
Therefore, the triangle ABC has angles A = 41.8 degrees, B = 77.5 degrees, and C = 60.7 degrees.