How do I know if my SSA triangle has more than one set of answers?

Question

Solve the triangle using the law of sines:

a = 75

b = 100

c = ?

 

<A = 30°

<B = ?

<C =?

 

My work:

Sin B / 100 = sin 30 / 75

Sin B = (100 * sin(30°))/75

Sin B = .6667

Sin^-1 (.6667) = B = 41.8°

therefore 180° - 30° - 41.8° = <C = 108.2°

 

c / Sin 108.2 = 100 / Sin 41.8

c = (Sin (108.2) * 100) / sin 41.8)

c = 142.5

 

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Now I know there is something that you're supposed to do to find a new degrees answer for <C such as <C = 180° - 41.8° = 138.2° as the new angle C and then calculate a new range of answers but I don't really understand it and to which SSA triangles it applies to

 

Answer 1

Another way of looking at it is this.

 

In triangle ABC, let sides a and b be given and also angle A.

 

Case 1: a < b sin A.

  No possible triangles.

 

Case 2: a > b sin A.

  Two triangles possible. They differ by mirroring side a about the altitude dropped from vertex C. Hence at least one of the solutions will be an obtuse triangle.

 

Case 3: a = b sin A.

  Unique solution. This case corresponds to the HL theorem for right triangles.

Answer 2

Basically any time you use the Law of Sines to find an angle, you could have 0, 1, or 2 possible triangles.  

 

If you get that the sin B > 1, then you get no possible triangles.

 

If you find an angle (as you did), you then want to check to see if the obtuse angle with the same reference angle can also work in your triangle.  This is because the acute angle you get from your calculator and the obtuse angle (180 - B) have the same value for sine, but your calculator will only give you the acute one.  (Or, sine is positive in the first and second quadrants if you guys have talked about that.)

 

So if <B = 41.8, then it might also be 180 - 41.8 = 138.2.  They both have a sine of .6667.

 

Once you find your angles, you try to see if you can make triangles with just one or possibly both of them.  Since the only other angle we know is 30 degrees, there's still enough room for a third angle using 138.2 and 30 (<C would be 11.8 degrees).  So in this case there are 2 triangles.

 

As an example where you just get 1 triangle: If you got 25 degrees for B, then 180 - 25 would be a 155 degree obtuse angle, and then the 30 + 155 would already be too much to have a third angle.

 

Overall, if the acute angle you find is larger than the acute angle you already have, then you can find an obtuse angle that will lead to a second triangle.