How do you solve the triangle given m∠B = 105°, b = 23, a = 14?

Start by drawing a diagram.

Since we have one angle with an opposite side and another side known, we are dealing with an ambiguous case. There are either #0, 1 or 2# solutions to this triangle.

We start by determining the measure of #A# using the law of Sines, since we already know the measure of side #a#.

#sinA/a = sinB/b#

#sinA/14 = (sin105˚)/23#

#A = arcsin((14sin105˚)/23)#

#A = 36˚#

There is only one solution to this triangle, because if it was, then the alternative measure of A would be #180˚ - 36˚ = 144˚#, which when added to #B#, would make the sum of the angles in the triangle exceed #180˚#.

We can now use the measure of angles #A and B# to solve for angle C.

#A + B + C = 180˚#

#36˚ + 105˚ + C = 180˚#

#C = 39˚#

The last step is using this information to reapply the law of sines and determine the length of side #c#.

#sinB/b = sinC/c#

#(sin105˚)/23 = (sin39˚)/c#

#c = (23sin39˚)/(sin105˚)#

#c= 15#

In summary

The triangle has the following measures.

#•A = 36˚#
#•a = 14" units"#
#•B = 105˚#
#•b = 23" units"#
#•C = 39˚#
#•c = 15" units"#

Hopefully this helps!