Trig question Please help!

Question

Two sides and an angle are given below. Determine whether the given information results in one​ triangle, two​ triangles, or no triangle at all. Solve any​ triangle(s) that results. b=8 c=9 and angle(B)=10 degrees

Answer 1

Using the Law of Sines, sin10° / 8 = sinC / 9

9sin10° = 8sinC

sinC = (9sin10°) / 8 = 0.195

The sine function is positive in quadrants 1 and 2, So, there are 2 possibilities for ∠C:

∠C = Sin^-1(0.195) or 180° - Sin^-1(0.195) = 11.2^° or 168.8°

There are two triangles:

First triangle: ∠B = 10°, ∠C = 11.2°, ∠A = 158.8°, b = 8, c = 9. Find side a by Law of Sines.

Second triangle: ∠B = 10°, ∠C = 168.8°, ∠A = 1.2°, b = 8, c = 9, Find side a using Law of Sines.

Answer 2

It depends on whether or not the question is asking if there is ANY triangle that can have those values or if it must be a right triangle. If you are looking for a right triangle, there is none, assuming that side b is opposite angle B. However, if you use Law of Sines, you can find angles C and A and side a, so this is a triangle. If you had a picture that goes with this information, it may change the answers.

Law of Sines:

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

To solve for angle C:

sin C = c (sin B)/ b, take inverse sin of both sides of the equation

Angle A comes from knowing every triangle three angles that sum to 180 degrees and you know two of the three angles

side a comes from solving for a using any of the other ratios in the Law of Sines, since you now have b, B, c, and C. I would recommend using the "b" ratio since that has the values you started with.

You should be able to do the math from here.