The Ambiguous Case: if a = 7,

Question

The Ambiguous Case: if a = 7, <A = 30°, and c = 12, how many distinct triangles can be created?

goformit100 · Answer

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anonymous · Answer

The answer are 2 triangles, but how do you determine that?

anonymous · Answer

$$\frac{\sin 30º}{7} = \frac{\sin C}{12}$$
12 sin 30º = 7sin C
6 = 7sin C
6/7 = sin C
<C = 58.997º or 121.003º

goformit100 · Answer

thats correct @Calcmathlete . and thankyou, as i didn't know to solve it in this way...

anonymous · Answer

Right, but from there, how were you able to determine two triangles can be formed? I understand how you got <C = 58.997 but how did you get 121?

anonymous · Answer

THe ambiguous case occurs when it's a Side-Side-Angle. If the included side is bigger than the other side, then it's either 2 or 0 triangles possible. To figure it out, when you get to the point: 6/7 = sin C, if it were a value greater then 1, it'd be 0. Since this is the law of sines, the value of sin is the same on 2 points of the unit circle. Subtract that angle from 180 to get it.

anonymous · Answer

For an example:

anonymous · Answer

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