Assume ๐›ผ is opposite side a, ๐›ฝ is opposite side b, and ๐›พ is opposite side c. Assume that a = 7sqrt(3), b = 7sqrt(2), ๐›ฝ = 45ยฐ. Determine whether there is no triangle, one triangle, or two triangles. Then solve each triangle, if possible. Enter your answers so that ๐›ผ1 is greater than ๐›ผ2.)

(To the extent possible use standard angles to simplify your responses. When necessary express your answers using the functions sin, cos, sinโˆ’1, cosโˆ’1. If there is no solution enter IMPOSSIBLE.)

2 answers

sin๐›ผ/7โˆš3 = sin45ยฐ/7โˆš2
now, ๐›พ = 180-๐›ผ-๐›ฝ
and you can use the law of sines/cosines to find c
You are looking at what is called the "ambiguous case"

Let's find angle ๐›ผ , using the sine law

sin๐›ผ/7โˆš3 = sin 45/7โˆš2
sin ๐›ผ = 7โˆš3(โˆš2/2) / 7โˆš2 = โˆš3/2
so ๐›ผ = 60ยฐ or ๐›ผ = 120ยฐ , 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 = 75ยฐ
find the missing side c using the sine law.

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