SOLUTION: Determine the measure of angle B of triangle ACB, given the following information: A=69degrees, a=25feet, b=16feet

This is called the "ambiguous case", when you 
are given two sides and an angle which is not 
included between them.  The ambiguous case has
0, 1, or 2 solutions.  

We use the law of sines:

   a          b          c
-------- = -------- = --------
 sin(A)     sin(B)     sin(C)
 
Here we only need:

   a          b          
-------- = -------- 
 sin(A)     sin(B)     
 
   25          16          
--------- = -------- 
 sin(69°)     sin(B)

Cross multiply:

25·sin(B) = 16·sin(69°)

Divide both sides by 25 to solve for sin(B)

             16·sin(69°)          
  sin(B) = ------------- 
                 25 

  sin(B) = .597491473

1. Sometimes this step comes our greater than 1.
   If so, there are no solutions. 

2. Sometimes this step comes out exactly 1. If so
   then B is a right angle and there is 1 solution.

3. Usually this step comes out a positive number
   between 0 and 1.  If this is the case, then there
   are either one or two solutions.  We must check to 
   see if there are two solutions.

   (a) Find the first quadrant solution for angle B

         sin(B) = .597491473

       Find the inverse sine of that right side:

              B = 36.6094481°

Since the sine is positive in the 2nd quadrant, there
is another solution for B, namely its supplement:

             B = 180 - 36.6094481° = 143.3095519°

Now we calculate what angle C would have to be if that
were a possible solution for angle B.

                A + B + C = 180°
   69° + 143.3095519° + C = 180°
         212.3095519° + C = 180°
                        C = 180° - 212.3095519
                        C = -32.3095519°

C cannot be a negative angle, so there is but
one solution, namely

              B = 36.6094481°

Since the sides are only given to two significant
digits, we should round the answer to the nearest 
degree:

             B = 37°

Edwin