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