State whether the given measurements determine zero, one, or two triangles. C = 37°, a = 16, c = 14

Answer 1

Angle C = 37, side a = 16 and side c = 14.

h = a * sin(C) = 16 * sin(37) = 9.629.

Necessary condition : h(=9.629) < c(=14) < a(=16), So the possible triangles are two.

By the Law of Sines in the reciprocal form

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

sin(A) = a * [sin(C) / c]

Substitute the values of C = 37, a = 16 and c = 14

sin(A) = (16) * [sin(37) / 14] = 0.6878.

A =  sin ^-1(0.6878) = 43.46⁰.

There are two angles , A₁ = 43.46⁰ and A₂ = 180 - 43.46⁰ = 136.54⁰,

 

For A₁ = 43.46⁰, we obtain

B = 180 - 37 - 43.46 = 99.54⁰.

The remaining side is

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

b = [c/sin(C)] * sin(B)

b = [14/sin(37)] * sin(99.54) = 22.94 units.

Angles A =43.46⁰, B = 99.54⁰ and side b = 22.94.

 

For A₂ = 136.54⁰, we obtain

B = 180 - 37 - 136.54 = 6.46⁰.

The remaining side is

b = [c/sin(C)] * sin(B)

b = [14/sin(37)] * sin(6.46) = 2.61 units.

Angles A =136.54⁰, B = 6.46⁰ and side b = 2.61.

Therefore,the two triangles are formed.

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