Answer:
A ≈ 25.6°, B ≈ 52.3°, C ≈ 68.8° or A
Explanation:
To solve the triangle ABC, we can use the Law of Cosines, which relates the lengths of the sides of a triangle to the cosine of the angles opposite those sides. The law states that:
c^2 = a^2 + b^2 - 2ab cos(C)
where c is the length of the side opposite angle C, a is the length of the side opposite angle A, and b is the length of the side opposite angle B.
Using the given values of a, b, and c, we can substitute into the formula and solve for cos(C):
41^2 = 19^2 + 45^2 - 2(19)(45) cos(C)
1681 = 361 + 2025 - 1710 cos(C)
cos(C) = (361 + 2025 - 1681) / (21945)
cos(C) = 0.37103
Now we can use the inverse cosine function to find the measure of angle C:
C = cos^-1(0.37103)
C ≈ 68.8°
Next, we can use the Law of Sines to find the measures of angles A and B:
sin(A)/a = sin(C)/c
sin(A)/19 = sin(68.8°)/41
sin(A) ≈ 0.4250
A ≈ 25.6°
Similarly, we can find the measure of angle B:
sin(B)/b = sin(C)/c
sin(B)/45 = sin(68.8°)/41
sin(B) ≈ 0.7873
B ≈ 52.3°
Therefore, the solution for the triangle ABC is:
A ≈ 25.6°, B ≈ 52.3°, C ≈ 68.8° or A