law of sines

Question

Two triangles can be formed with the given information. Use the Law of Sines to solve the triangles.  A = 55°, a = 12, b = 14

Answer 1

To form two triangles, draw the geometric diagram.

Law of sines : a/sin A = b/sin B = c/sin C.

Given : Angle A = 55⁰ and sides a = 12, b = 14.

Lets find remaining angles B and C and the remaining side c.

Law of sines : a/sin A = b/sin B ⇒ sin B = (b * sin A)/a.

sin B = (14 * sin 55⁰)/12 = (14 * 0.8191)/12 = 0.9556

⇒ B = sin^{- 1}(0.9556) = 72.86⁰ .

There are two triangles, B = 72.86⁰ and B₁ = 180 - 72.86 = 107.14⁰  [Since sin(72.86⁰) = sin(107.14⁰) = 0.9556]

Find side c for angle B.

In triangle ABC, sum of angles = 180⁰ .

A + B + C = 180⁰

55⁰ + 72.86⁰ + C = 180⁰

C = 180⁰ - 127.86⁰

⇒ C = 52.14⁰ .

Law of sines : b/sin B = c/sin C ⇒ c = (b * sin C)/sin B.

⇒ c = (14 * sin 52.14⁰)/sin 72.86⁰ = (14 * 0.7895)/0.9556 = 11.053/0.9556 = 11.566.

 

Find side c for angle B₁.

In triangle ABC, sum of angles = 180⁰ .

A + B₁ + C = 180⁰

55⁰ + 107.14⁰ + C = 180⁰

C = 180⁰ - 162.14⁰

⇒ C = 17.86⁰ .

Law of sines : b/sin B = c/sin C ⇒ c = (b * sin C)/sin B.

⇒ c = (14 * sin 17.86⁰)/sin 107.14⁰ = (14 * 0.3067)/0.9556 = 4.2938/0.9556 = 4.4933.