Solve the triangle ABC under the given conditions.

a = 19, b = 8.2, c = 12

A=
B=
C=

To solve the triangle ABC, we will use the Law of Cosines and the Law of Sines. The Law of Cosines states that in any triangle ABC with sides a, b, and c, and angles A, B, and C opposite the corresponding sides, the following formula holds:

c^2 = a^2 + b^2 - 2ab * cos(C)

We can use this formula to find angle C. Plugging in the known values, we have:

12^2 = 19^2 + 8.2^2 - 2 * 19 * 8.2 * cos(C)

Simplifying further, we get:

144 = 361 + 67.24 - 313.6 * cos(C)

Rearranging the equation, we have:

313.6 * cos(C) = 361 + 67.24 - 144

313.6 * cos(C) = 284.24

cos(C) = 284.24 / 313.6

Using a calculator, we find:

cos(C) ≈ 0.905

To find the angle C, we take the inverse cosine of 0.905:

C ≈ arccos(0.905)

Using a calculator, we find:

C ≈ 25.805°

Now, to find the remaining angles A and B, we can use the Law of Sines. The Law of Sines states that in any triangle ABC, the following ratio holds:

sin(A) / a = sin(B) / b = sin(C) / c

Using this formula, we can find angles A and B. Plugging in the known values, we have:

sin(A) / 19 = sin(25.805°) / 12

Cross multiplying, we get:

sin(A) = (19 * sin(25.805°)) / 12

Using a calculator, we find:

sin(A) ≈ 0.409

Taking the inverse sine of 0.409, we find:

A ≈ arcsin(0.409)

Using a calculator, we find:

A ≈ 24.466°

Similarly, we can find angle B using the Law of Sines:

sin(B) / 8.2 = sin(25.805°) / 12

Cross multiplying, we get:

sin(B) = (8.2 * sin(25.805°)) / 12

Using a calculator, we find:

sin(B) ≈ 0.251

Taking the inverse sine of 0.251, we find:

B ≈ arcsin(0.251)

Using a calculator, we find:

B ≈ 14.73°

Therefore, the solution to the triangle ABC is as follows:

A ≈ 24.466°
B ≈ 14.73°
C ≈ 25.805°