Question

The given measurements may or may not determine a triangle. If not, then state that no triangle is formed. If a triangle is formed, then use the Law of Sines to solve the triangle, if it is possible, or state that the Law of Sines cannot be used. B = 126°, c = 7, b = 12 (2 points) C = 25.8°, A = 28.2°, a ≈6.5 C = 28.2°, A = 25.8°, a ≈6.5 No triangle is formed. The triangle cannot be solved with the Law of Sines.

Answer 1

Answer:

C = 28.2, A = 25.8, a = 6.5

See diagram below.

==============================================

Work Shown:

Given info is

B = 126 degrees

b = 12

c = 7

Use the Law of Sines to solve for angle C

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

sin(C)/7 = sin(126)/12

sin(C)/7 = 0.067418082864579

sin(C) = 7*0.067418082864579

sin(C) = 0.471926580052053

C = arcsin(0.471926580052053) or C = 180-arcsin(0.471926580052053)

C = 28.1594278560921 or C = 180-28.1594278560921

C = 28.1594278560921 or C = 151.840572143908

C = 28.2 or C = 151.8

We have two possible angle values for C.

-----------------

If C = 28.2, then A = 180-B-C = 180-126-28.2 = 25.8

If C = 151.8, then A = 180-B-C = 180-126-151.8 = -97.8

So it is not possible for C = 151.8 (because it leads to angle A being negative)

Therefore, only C = 28.2 is possible.

-----------------

Use the law of cosines to find the remaining side 'a'

a^2 = b^2 + c^2 - 2*b*c*cos(A)

a^2 = (12)^2 + (7)^2 - 2*(12)*(7)*cos(25.8)

a^2 = 144 + 49 - 168*0.900318771402194

a^2 = 144 + 49 - 151.253553595569

a^2 = 41.7464464044315

a = sqrt(41.7464464044315)

a = 6.46114900032738

a = 6.5

-----------------

Only one triangle is possible

The fully solved triangle has these angles and sides:

• A = 25.8 (approx)
• B = 126
• C = 28.2 (approx)
• a = 6.5 (approx)
• b = 12
• c = 7

With stuff in bold representing the terms we solved for previously. Attached below is an image of the fully solved triangle.