sin๐ผ/7โ3 = sin45ยฐ/7โ2
now, ๐พ = 180-๐ผ-๐ฝ
and you can use the law of sines/cosines to find c
Assume ๐ผ is opposite side a, ๐ฝ is opposite side b, and ๐พ is opposite side c. Assume that a = 7sqrt(3), b = 7sqrt(2), ๐ฝ = 45ยฐ. Determine whether there is no triangle, one triangle, or two triangles. Then solve each triangle, if possible. Enter your answers so that ๐ผ1 is greater than ๐ผ2.)
(To the extent possible use standard angles to simplify your responses. When necessary express your answers using the functions sin, cos, sinโ1, cosโ1. If there is no solution enter IMPOSSIBLE.)
2 answers
You are looking at what is called the "ambiguous case"
Let's find angle ๐ผ , using the sine law
sin๐ผ/7โ3 = sin 45/7โ2
sin ๐ผ = 7โ3(โ2/2) / 7โ2 = โ3/2
so ๐ผ = 60ยฐ or ๐ผ = 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.
Let's find angle ๐ผ , using the sine law
sin๐ผ/7โ3 = sin 45/7โ2
sin ๐ผ = 7โ3(โ2/2) / 7โ2 = โ3/2
so ๐ผ = 60ยฐ or ๐ผ = 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.