Ambiguous case of the law of sines

The ambiguous case of the law of sines happens when two sides and an angle opposite one of them is given. We can shorten this situation with SSA.

Since the length of the third side is not known, we don't know if a triangle will be formed or not. That is the reason we call this case ambiguous.

In fact, this kind of situation or SSA can give the following 4 scenarios.

  • No triangle
  • 1 triangle
  • 1 right triangle
  • 2 triangles

First scenario of the ambiguous case of the law of sines : a < h

For example, look at the triangle below where a, b, and angle A are given.

Ambiguous case of the law of sines: no triangleAmbiguous case of the law of sines: no triangle

Because a is shorter than h, a is not long enough to form a triangle. In fact, the number of possible triangles that can be formed in the SSA case depends on the length of the altitude or h.

Notice that sin A =
h / b

Multiply both sides by b and we get h = b sin A

Example: suppose A = 74°, a = 51, and b = 72.

h = 72 × sin (74°) = 72 × 0.9612 = 68.20

Since 51 or a is less than h or 69.20, no triangle will be formed.

Second situation:  a = h

When a  = h, the resulting triangle will always be a right triangle.

Ambiguous case of the law of sines: right triangleRight triangle

Example: suppose A = 30, a = 25, and b = 50.

h = 50 × sin (30°) = 50 × 0.5 = 25

Since 25 or a is equal to h or 25, 1 right triangle will be formed.

Third situation:  a > h and a > b

When a is bigger than h, again a triangle can be formed. However, since a is bigger than b, we can only have one triangle. Try to make a triangle where a is bigger than b, you will notice that there can only be 1 such triangle.

Ambiguous case of the law of sines: 1 triangle1 triangle

Example: suppose A = 30, a = 50, and b = 40.

h = 40 × sin (30°) = 40 × 0.5 = 20

Since 50 or a is bigger than both h or 20 and b or 40, 1 triangle will be formed.

Last situation:  a > h and a < b

When a is less than b, 2 triangles can be formed as clearly illustrated below. The two triangles are triangle ACD and triangle AED. 

Ambiguous case of the law of sines: 2 triangles2 triangles

Example: suppose A = 30, a = 40, and b = 60

h = 60 × sin (30°) = 60 × 0.5 = 30

Since 40 or a is bigger than h and a is smaller than b or 60, 2 triangles will be formed.

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