Answer:
Two possible solutions
Step-by-step explanation:
we know that
Applying the law of sines
[tex]\frac{a}{sin(A)}=\frac{b}{Sin(B)}=\frac{c}{Sin(C)}[/tex]
we have
[tex]a=32\ units[/tex]
[tex]b=27\ units[/tex]
[tex]B=37\°[/tex]
step 1
Find the measure of angle A
[tex]\frac{a}{sin(A)}=\frac{b}{Sin(B)}[/tex]
substitute the values
[tex]\frac{32}{sin(A)}=\frac{27}{Sin(37\°)}[/tex]
[tex]sin(A)=(32)Sin(37\°)/27=0.71326[/tex]
[tex]A=arcsin(0.71326)=45.5\°[/tex]
The measure of angle A could have two measures
the first measure-------> [tex]A=45.5\°[/tex]
the second measure -----> [tex]A=180\°-45.5\°=134.5\°[/tex]
step 2
Find the first measure of angle C
Remember that the sum of the internal angles of a triangle must be equal to [tex]180\°[/tex]
[tex]A+B+C=180\°[/tex]
substitute the values
[tex]A=45.5\°[/tex]
[tex]B=37\°[/tex]
[tex]45.5\°+37\°+C=180\°[/tex]
[tex]C=180\°-(45.5\°+37\°)=97.5\°[/tex]
step 3
Find the first length of side c
[tex]\frac{a}{sin(A)}=\frac{c}{Sin(C)}[/tex]
substitute the values
[tex]\frac{32}{sin(37\°)}=\frac{c}{Sin(97.5\°)}[/tex]
[tex]c=Sin(97.5\°)\frac{32}{sin(37\°)}=52.7\ units[/tex]
therefore
the measures for the first solution of the triangle are
[tex]A=45.5\°[/tex] , [tex]a=32\ units[/tex]
[tex]B=37\°[/tex] , [tex]b=27\ units[/tex]
[tex]C=97.5\°[/tex] , [tex]b=52.7\ units[/tex]
step 4
Find the second measure of angle C with the second measure of angle A
Remember that the sum of the internal angles of a triangle must be equal to [tex]180\°[/tex]
[tex]A+B+C=180\°[/tex]
substitute the values
[tex]A=134.5\°[/tex]
[tex]B=37\°[/tex]
[tex]134.5\°+37\°+C=180\°[/tex]
[tex]C=180\°-(134.5\°+37\°)=8.5\°[/tex]
step 5
Find the second length of side c
[tex]\frac{a}{sin(A)}=\frac{c}{Sin(C)}[/tex]
substitute the values
[tex]\frac{32}{sin(37\°)}=\frac{c}{Sin(8.5\°)}[/tex]
[tex]c=Sin(8.5\°)\frac{32}{sin(37\°)}=7.9\ units[/tex]
therefore
the measures for the second solution of the triangle are
[tex]A=45.5\°[/tex] , [tex]a=32\ units[/tex]
[tex]B=37\°[/tex] , [tex]b=27\ units[/tex]
[tex]C=8.5\°[/tex] , [tex]b=7.9\ units[/tex]
