Question

Assume 𝛼 is opposite side a, 𝛽 is opposite side b, and 𝛾 is opposite side c. Assume that a = 7sqrt(3), b = 7sqrt(2), 𝛽 = 45°. Determine whether there is no triangle, one triangle, or two triangles. Then solve each triangle, if possible. Enter your answers so that 𝛼1 is greater than 𝛼2.)
(To the extent possible use standard angles to simplify your responses. When necessary express your answers using the functions sin, cos, sin−1, cos−1. If there is no solution enter IMPOSSIBLE.)

Answer 1

sin𝛼/7√3 = sin45°/7√2
now, 𝛾 = 180-𝛼-𝛽
and you can use the law of sines/cosines to find c

Answer 2

You are looking at what is called the "ambiguous case"

Let's find angle 𝛼 , using the sine law

sin𝛼/7√3 = sin 45/7√2
sin 𝛼 = 7√3(√2/2) / 7√2 = √3/2
so 𝛼 = 60° or 𝛼 = 120° , since the sine is positive in both the 1st and 2nd quadrants.

So it looks like we could have 2 triangles.
Triangle #1, angles B = 45, angle A = 60 and angle C = 75°
find the missing side c using the sine law.

triangle #2, angles 45, 120 and 15° , again find the missing side.
Side c should be the same for both.