What is this asking and how do I solve it?

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 ∠B₁ is smaller than ∠B₂.)

a = 78,    b = 104,    ∠A = 27°

∠B1 =	°		∠B2 =	°
∠C1 =	°		∠C2 =	°
c1 =			c2 =

Answer 1

a < h, none

a = h, one

a ≥ c, none

h < a < c, two

sin 27 = h / 104

0.454 = h / 104

47.22 = h

There are two, sketch and use Law of Sines.

Answer 2

Imagine a triangle where the sides are of length a, b, and c, and the angles are A, B, and C. Angle A is opposite side a, angle B is opposite side b, and angle C is opposite side c.

The law of sines for such a triangle is as follows:

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

We are given:

a = 78

b = 104

A = 27°

Using the Law of Sines, we can solve for angle B:

78/ (sin 27°) = 104/ (sin B)

sin B = [104 * sin 27°] / 78

sin B = 0.61

B = sin^-1 (0.61) = 37°

This is one solution for angle B. There are actually two angles equivalent to sin^-1(0.61), that are complementary to each other (they add to 180°). The calculator will give you the answer that is less than 90°, so to find the other solution, we can take 180 - 37 = 143°.

So angle B1 is 37° and angle B2 is 143°.

If angle A is 27°, and angle B1 is 37°, then angle C1 = 180° - 27° - 37° = 116°.

If angle A is 27°, and angle B2 is 143°, then angle C2 = 180° - 27° - 143° = 10°.

Now we can solve for side c1:

78 / (sin 27°) = c1 / (sin 116°)

c1 = 78*(sin 116°) / (sin 27°) = 154

Solving for side c2:

78 / (sin 27°) = c2 / (sin 10°)

c2 = 78*(sin 10°) / (sin 27°) = 30