Two sides and an angle

Question

Two sides and an angle are given below. Determine whether the given information results in one​ triangle, two​ triangles, or no triangle at all. Solve any resulting​ triangle(s).

b=6, c=7, B=30°

Answer 1

°This is an ambiguous case. For two triangle solution, the following conditions have to hold:

1) The given must be in this order: Side-Side-Angle (SSA) either going clockwise or counterclockwise.

h = height of the triangle.

a, b, c are the sides.

∠B is one of its angles.

2) h < b < c

h = c sin 30º

h = 7 (0.5) = 3.5

3.5 < 6 < 7

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Since it satisfies all the conditions, therefore we have two triangle solutions.

ΔABC₁ and ΔABC₂

1)For ΔABC₂, use Law of Sine:

b / sin B = c / sin C

6 / sin 30º = 7 / sin C

sin C = 7*sin 30º / 6

m∠C = sin^-1(7*sin 30º / 6)

m∠C₂ ≈ 35.69º

The sum of all angles is 180 degrees:

m∠A = 180 -m∠B - m∠C

m∠A = 180 - 30 - 35.69

m∠A ≈ 114.31°

For the third side (A), Law of Sine again:

a / sin A = b / sin B

a = 6*sin 114.31°/ sin 30°

a ≈ 10.94

2)For ΔABC₁, Solve for angle C₁ first:

m∠C = 180 -35.69

m∠C₁ ≈ 144.31°

m∠A ≈ 180 - 30 - 144.31

m∠A ≈ 5.69°

a / sin A = b / sin B

a = 6* sin 5.69° / sin 30°

a ≈ 1.19