Atriangle has sides a, b, and c. the angle between sides a and b is #pi/6# and the angle between sides b and c is #pi/12#. if side b has a length of 5, what is the area of the triangle?

Use the Law of Sines to solve for side c. Then you can you the two sides (c and b) and their included angle to find the area using.
sin(pi/6)/c = pi/12/5   cross multiply and divide 5*sin(pi/6)/pi/12 = 9.5
That makes side c= 9.5.
now use the formula for finding area with two sides and their included angle. It must the angle between them.
(1/2)(9.5)(5)cos(pi/12) multiply that all together and you should get the area equals 22.9