Trigonometry Problem

Question

Two sides and an angle ( SSA ) of a triangle are given . Determine whether the given measurements produce one triangle Two triangles , or no triangle at all . Solve each triangle that results . Round lengths to the nearest tenth and angle measures to the nearest degree B = 26 degrees b = 11.1 . a = 12.66

Answer 1

Use the Law of Sines:

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

We are given

a = 12.66

b = 11.1

B = 26^o

sin A / a = sin B / b leads to

sin A / 12.66 = sin(26^o) / 11.1

sin(A) = 12.66 * sin(26^o) / 11.1 = 0.5000

A = 30^o or 150^o

If A = 30^o, then C = 180 - A - B = 180 - 30 - 26 = 124^o

c = sin C * b / sin B = sin(124^o) * 11.1 / sin(26^o) = 20.9920

if A = 150, then C = 180 - A - B = 180 - 150 - 26 = 4^o

c = sin C * b / sin B = sin(4^o) * 11.1 / sin(26^o) = 1.7663

Answer: 2 Solutions --

A, B, C = 30, 26, 124 degrees

a, b, c = 12.7, 11,1, 21.0

A, B, C = 150, 26, 4 degrees

a, b, c = 12.7, 11,1, 1.8

Answer 2

Here's a picture of what you have described:

You can see that when the side opposite the angle (side b) is shorter than the side adjacent to the angle (side a), then there are two ways you can flip side b to make a triangle. You can make angle C be very small as shown with the green, or you can make angle C be obtuse as shown in red.