Answer:
Part 1) [tex]C=39.3\°[/tex]
Part 2) [tex]A=68.7\°[/tex]
Part 3) [tex]a=11.8\ units[/tex]
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]
In this problem we have
[tex]B=72\°[/tex]
[tex]b=12\ units[/tex]
[tex]c=8\ units[/tex]
Step 1
Find the measure angle C
[tex]\frac{b}{sin(B)} =\frac{c}{sin(C)}[/tex]
substitute the values and solve for C
[tex]\frac{12}{sin(72\°)} =\frac{8}{sin(C)}\\ \\sin(C)=sin(72\°)*(8/12)\\ \\sin(C)= 0.6340\\ \\C=39.3\°[/tex]
Step 2
Find the measure of angle A
we know that
The sum of the internal angles of the triangle must be equal to [tex]180[/tex] degrees
so
[tex]m<A+m<B+m<C=180\°[/tex]
we have
[tex]m<B=72\°[/tex]
[tex]m<C=39.3\°[/tex]
[tex]m<A+72\°+39.3\°=180\°[/tex]
[tex]m<A=180\°-72\°-39.3\°=68.7\°[/tex]
Step 3
Find the measure of side a
Applying the law of cosines
[tex]a^{2}=b^{2}+c^{2}-2(b)(c)cos(A)[/tex]
substitute
[tex]a^{2}=12^{2}+8^{2}-2(12)(8)cos(68.7\°)[/tex]
[tex]a^{2}=208-(192)cos(68.7\°)[/tex]
[tex]a^{2}=138.26[/tex]
[tex]a=11.8\ units[/tex]
