Law of sines question

Question

Use the Law of Sines to solve for all possible triangles that satisfy the given conditions. (If an answer does not exist, enter DNE. Round your answers to one decimal place. Below, enter your answers so that ∠A₁ is smaller than ∠A₂.)

b = 27,    c = 38,    ∠B = 28°

Answer 1

sinC / 38 = sin28°/27

27sinC = 38sin28°

sinC = 0.6607

∠C = Sin^-1(0.6607) or 180° - Sin^-1(0.6607) = 41.4° or 138.6°

If ∠C = 41.4°, then ∠A = 180° - 28° - 41.4° = 110.6°

So, sin110.6° / a = sin28°/27. Therefore, a = 53.8

If ∠C = 138.6°, then ∠A = 180° - 138.6° - 28° = 13.4°

So, sin13.4° / a = sin28° / 27 Therefore, a = 13.3

First Triangle: a = 53.8, b = 27, c = 38, ∠A = 110.6°, ∠B = 28°, ∠C = 41.4°

Second Triangle: a = 13.3, b = 27, c = 38, ∠A = 13.4°, ∠B = 28°, ∠C = 138.6°