Question

Solve the triangle. The Law of Cosines may be needed. (Standard notation for triangle ABC is used throughout. Use a calculator and give your answers as decimals to tenth at the end of the computation. If no such triangle exists, enter 'NONE' in each of the answer blanks.)

b = 31.9, c = 14.7, C = 27.8°

Answer 1

Explanation:

To solve the triangle, we can use the Law of Cosines and the fact that the sum of angles in a triangle is 180 degrees.

Given:

b = 31.9

c = 14.7

C = 27.8°

To find angle A, we can use the Law of Cosines:

a^2 = b^2 + c^2 - 2bc*cos(A)

We know b, c, and C, so we can substitute the values:

a^2 = 31.9^2 + 14.7^2 - 2 * 31.9 * 14.7 * cos(27.8°)

Now, we can solve for a:

a = sqrt(31.9^2 + 14.7^2 - 2 * 31.9 * 14.7 * cos(27.8°))

To find angle B, we can use the Law of Sines:

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

We know b and a, so we can substitute the values:

sin(B)/31.9 = sin(A)/a

Now, we can solve for sin(B):

sin(B) = (sin(A)/a) * 31.9

Finally, we can solve for angle B:

B = arcsin((sin(A)/a) * 31.9)

To find angle C, we can subtract angles A and B from 180°:

C = 180° - A - B

Now, we can calculate the values:

A = acos((b^2 + c^2 - a^2) / (2 * b * c))

B = arcsin((sin(A)/a) * 31.9)

C = 180° - A - B

Using the given values, we can calculate the triangle:

A ≈ acos((31.9^2 + 14.7^2 - a^2) / (2 * 31.9 * 14.7))

B ≈ arcsin((sin(A)/a) * 31.9)

C ≈ 180° - A - B

Please note that the final numerical calculation is required to provide the exact values for A, B, and C.