Question

Trig.. trying to determine how many triangles and find the missing sides/ angles... b=5, c=3, B = 100 .. Help please

Answer 1

This is the Ambiguous Case in solving triangles because you are given two sides and the non-included angle and asked to find the other angles and sides of the triangle.

Using the Law of Sines,
Sine C /3 = Sine 100/5
Sine C / 3 = .985 / 5
Sine C = [.985 (3)] / 5
Sine C = .5908
C = 36.22 degrees OR 180 - 36.22 = 143.78 degrees

If C were 143.78 degrees, it and the 100 degrees of angle B could not be in the same triangle.
Therefore, C = 36.22 degrees is the only viable option for C and there is only one possible triangle.

In that one triangle, angle A has measure 43.78 degrees [180 - 36.22 - 100]
Continuing with the Law of Sines,
Sine A / a = = Sine 100 / 5
Sine 43.78 / a = .985 / 5
.692 / a = .985 / 5
.985a = 5 ( .692 )
a = 3.513

Angles Sides
A: 43.78° a = 3.513
B: 100° b = 5
C: 36.22° c = 3