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.
For example, look at the triangle below where a, b, and angle A are given.
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.
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.
When a = h, the resulting triangle will always be a right 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.
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.
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.
When a is less than b, 2 triangles can be formed as clearly illustrated below. The two triangles are triangle ACD and triangle AED.
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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