Question 1093693
law of sines says:


a / sin(A) = b / sin(B) = c / sin(C)


you are given:


B = 64 degrees
b = 22
a = 23.9


a / sin(A) = b / sin(B) becomes 23.9 / sin(A) = 22 / sin(64)


solve for sin(A) to get sin(A) = 23.9 * sin(64) / 22 = .9764


solve for A to get A = arcsin(.9764) = 77.5 degrees.


C is equal to 180 - 77.5 - 64 = 38.5 degrees.


c / sin(C) = b / sin(B) becomes c / sin(38.5) = 22 / sin(64)


solve for c to get c = 22 * sin(38.5) / sin(64) = 15.2


one solution is:


a. ∠A = 77.5°, ∠C = 38.5°, c = 15.2 mi


since the given angle B is not included between the given sides a and b, then there is the possibility of another solution.


that solution could be in the second quadrant.


check for a solution in the second quadrant as follows:


180 - 77.5 = 102.5 in the second quadrant.
that means that A is possibly 102.5
if possible, C would be equal to 180 - 102.5 - 64 = 13.5
that's possible so there is a second possible solution.


if you tried to do the same thing with 38.5, then you would have found it is not possible.
180 - 38.5 = 141.5 + 64 = something greater than 180, therefore not possible.


solve for c as follows:


c / sin(C) = b / sin(B) which becomes c/sin(13.5) = 22/sin(64).


solve for c to get c = 22 * sin(13.5) / sin(64) = 5.7


your second possible solution is:


b. ∠A = 102.5°, ∠C = 13.5°, c = 5.7 mi


since both a. and b. are possible, and they are both included in c., then your solution is:


c. ∠A = 77.5°, ∠C = 38.5°, c = 15.2 mi or ∠A = 102.5°, ∠C = 13.5°, c = 5.7 m


a reasonably scaled diagram of your solutions is shown below:


<img src = "http://theo.x10hosting.com/2017/091511.jpg" alt="$$$" >


in this diagram:


angle A is on top.
Angle B is lower left.
Angle C is lower right.