Q:

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

Accepted Solution

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