Question 1010678
In triangle ABC, measure of angle A is 32 degrees, AB equals 8 and BC 
equals 5.  Find the 2 possible values of measure angle C.
<pre>
{{{drawing(400,4200/19,-1,10.4,-1,5.3,
locate(0,0,A),locate(6.7,4.6,B),locate(4,0,C),green(locate(9.4,0,C)),
triangle(0,0,4.133370492,0,6.784384769,4.239354114),
line(4.133370492,0,6.784384769,4.239354114),
locate(.8,.5,"32°"), red(arc(0,0,3.4,-3.4,0,32)),
locate(3,2.48,8),locate(5,2.3,5),locate(3.8,.7,"?°"),
red(arc(4.133370492,0,2,-2,58,180)),
green(triangle(4.133370492,0,9.435399046,0,6.784384769,4.239354114),
arc(9.435399046,0,2,-2,122,180),locate(8.28,2.3,5),locate(8.7,.5,"?°")) )}}}

By the law of sines:

{{{BC/sin("BAC")}}}{{{""=""}}}{{{AB/sin("ACB")}}}

Cross-multiply

{{{BC*sin("ACB")}}}{{{""=""}}}{{{AB*sin("BAC")}}}

{{{sin("ACB")}}}{{{""=""}}}{{{AB*sin("BAC")/BC}}}

{{{sin("ACB")}}}{{{""=""}}}{{{8*sin("32°")/5}}}

{{{sin("ACB")}}}{{{""=""}}}{{{8*sin("32°")/5}}}

{{{sin("ACB")}}}{{{""=""}}}{{{0.8478708228}}}

Acute angles are QI angles and obtuse angles are
QII angles.  Since QI and QII angles have positive 
sines, there are two solutions for &#8736;ACB:

The acute angle C = &#8736;ACB can be found by using the 
inverse sine function on the calculator.  The
green acute angle is

<font color="green"><b>&#8736;ACB = 57.98083803°</b></font>

The black obtuse angle indicated by the red arc that has 
that same sine can be found by subtracting the green one
from 180°, for the green one is its referent angle.

&#8736;ACB = 180° - 57.98083803° = 122.019162°

Edwin</pre>