Question 34008
<pre><b><font size = 4>Solve this triangle: a=8.4 b=22.1 A=33°46'

                 
                 .C
  b = 22.1   .    \
         .         \a = 8.4
     .              \
 A.   33°46'         \B  
  ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯

This is the ambiguous case.  Sometimes there is no solution,
sometimes there is one solution, and sometimes there are two solutions.
There is no solution when a is too short to reach down to AB.

If the picture above shows one possible solution, here is another:

                 .C
  b = 22.1   .  / 
         .     /a = 8.4
     .        /     
 A.   33°46' / B  
  ¯¯¯¯¯¯¯¯¯¯¯

The side a can slant right or left if it is long enough to reach 
down to AB.  Sometimes a is so long that it can only slant to the right,
and if we slant it to the left it extends left of A, and not create a 
second solution.  That would be the case when there would be only one
solution.  We don't yet know how many solutions, if any, we have.

In any case we use the law of sines:
  
    a       b       c
   ———— = ———— = —————
   sinA   sinB    sinC

We only need this part of the law of sines:

    a       b      
   ———— = ———— 
   sinA   sinB   
    
We solve for sinB

   a·sinB = b·sinA

Divide both sides by sinB

     sinB = b·sinA/a

     sinB = (22.1)sin(33°46')/8.4

     sinB = 1.4623151
  
Uh oh!  The sine of an angle cannot exceed 1; therefore
there is no solution.  That means side a, which is 8.4,
is too short to reach down to AB, and we cannot draw a 
triangle with those parts.

Edwin
AnlytcPhil@aol.com</pre>