Answer:
The measures of the triangle are
[tex]A=32\°,B=23\°,C=125\°[/tex]
[tex]a=19\ units, b=14\ units, c=29.4\ units[/tex]
Step-by-step explanation:
Step 1
Find the value of angle B
Applying the law of sines
[tex]\frac{a}{sin(A)} =\frac{b}{sin(B)}[/tex]
we have
[tex]a=19\ units, b=14\ units, A=32\°[/tex]
Substitute and solve for B
[tex]\frac{19}{sin(32\°)} =\frac{14}{sin(B)}[/tex]
[tex]\frac{19}{sin(32\°)} =\frac{14}{sin(B)}\\ \\sin(B)=(14/19)*sin(32\°)[/tex]
[tex]B=arcsin((14/19)*sin(32\°))=23\°[/tex]
Step 2
Find the measure of angle C
we know that
The sum of the interior angles of triangle is equal to [tex]180\°[/tex]
so
[tex]A+B+C=180\°[/tex]
we have
[tex]A=32\°[/tex]
[tex]B=23\°[/tex]
Substitute and solve for C
[tex]32\°+23\°+C=180\°[/tex]
[tex]C=180\°-55\°=125\°[/tex]
Step 3
Find the length of the side c
Applying the law of cosines
[tex]c^{2} =a^{2}+b^{2}-2abcos(C)[/tex]
we have
[tex]a=19\ units, b=14\ units, C=125\°[/tex]
substitute
[tex]c^{2} =19^{2}+14^{2}-2(19)(14)cos(125\°)[/tex]
[tex]c^{2} =557-532cos(125\°)[/tex]
[tex]c =29.4\ units[/tex]
The measures of the triangle are
[tex]A=32\°,B=23\°,C=125\°[/tex]
[tex]a=19\ units, b=14\ units, c=29.4\ units[/tex]
