Determine the number of possible solutions for a triangle with B=37 degrees, a=32, b=27

Answer:Two possible solutionsStep-by-step explanation:we know thatApplying 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 1Find 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 measuresthe first measure-------> [tex]A=45.5\°[/tex] the second measure -----> [tex]A=180\°-45.5\°=134.5\°[/tex] step 2Find the first measure of angle CRemember 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 3Find 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]thereforethe 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 ARemember 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 5Find 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]thereforethe 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]