Solve △ABC, given a = 4.4, b = 9.1, and c = 5.7. Round your answer to the nearest tenth.
What do you mean by "solve"? You already have provided the three side lengths.
The three angles A, B, and C can be computed by using the law of cosines.
c^2 = a^2 + b^2 - 2 a *b cos C
cos C = (a^2 + b^2 - c^2)/(2 a b)
and two other rearrangements of the same formula for A and B
To solve the triangle △ABC, we can use the Law of Cosines, which states that in a triangle with sides a, b, and c, and angles A, B, and C respectively, the following equation holds:
c^2 = a^2 + b^2 - 2ab*cos(C)
Given the values a = 4.4, b = 9.1, and c = 5.7, we can substitute them into the equation:
(5.7)^2 = (4.4)^2 + (9.1)^2 - 2(4.4)(9.1)*cos(C)
Now, we can solve for cos(C):
cos(C) = [(4.4)^2 + (9.1)^2 - (5.7)^2] / [2(4.4)(9.1)]
cos(C) = (19.36 + 82.81 - 32.49) / (79.04)
cos(C) = 69.68 / 79.04
cos(C) ≈ 0.881
To find the angle C, we can take the inverse cosine (or arccosine) of 0.881:
C = arccos(0.881)
Using a calculator, we can find that C ≈ 29.0 degrees.
Now, to find angle A, we can use the Law of Sines, which states that in a triangle with sides a, b, and c, and angles A, B, and C respectively, the following equation holds:
sin(A) / a = sin(C) / c
Substituting the known values:
sin(A) / 4.4 = sin(29.0) / 5.7
Now, we can solve for sin(A):
sin(A) = (4.4 * sin(29.0)) / 5.7
Using a calculator, we can calculate sin(A) ≈ 0.415
Taking the inverse sine (or arcsine) of 0.415, we find that A ≈ 24.4 degrees.
To find angle B, we can use the fact that the sum of the interior angles of a triangle is 180 degrees:
B = 180 - A - C
B ≈ 126.6 degrees.
Therefore, the triangle △ABC has angles A ≈ 24.4 degrees, B ≈ 126.6 degrees, and C ≈ 29.0 degrees.