Answer:
First triangle
[tex]B=67.4\°[/tex]
[tex]C=53.6\°[/tex]
[tex]c=12.2\ units[/tex]
Second triangle
[tex]B=112.6\°[/tex]
[tex]C=8.4\°[/tex]
[tex]c=2.2\ units[/tex]
Step-by-step explanation:
In this problem we have
[tex]A=59\°[/tex]
[tex]a=13\ units[/tex]
[tex]b=14\ units[/tex]
First Triangle
Step 1
Find the value of angle B
Applying the law of sines
[tex]\frac{a}{sin(A)} =\frac{b}{sin(B)}[/tex]
substitute and solve for B
[tex]\frac{13}{sin(59\°)} =\frac{14}{sin(B)}\\ \\sin(B)=14*sin( 59\°)/13\\ \\sin(B)=0.9231\\ \\B=arcsin(0.9231)\\ \\B=67.4\°[/tex]
There are two measures of angle B, supplementary to each other
Step 2
Find the value of angle C
Remember that
the sum of the internal angles of a triangle is equal to [tex]180\°[/tex]
so
[tex]A+B+C=180\°[/tex]
we have
[tex]A=59\°[/tex]
[tex]B=67.4\°[/tex]
substitute and solve for C
[tex]59\°+67.4\°+C=180\°[/tex]
[tex]C=180\°-(59\°+67.4\°)=53.6\°[/tex]
Step 3
Find the measure of side c
Applying the law of sines
[tex]\frac{a}{sin(A)} =\frac{c}{sin(C)}[/tex]
substitute and solve for c
[tex]\frac{13}{sin(59\°)} =\frac{c}{sin(53.6\°)}[/tex]
[tex]c=\frac{13}{sin(59\°)}*sin(53.6\°)\\\\c=12.2\ units[/tex]
Second Triangle
Step 1
Find the value of angle B
Applying the law of sines
[tex]\frac{a}{sin(A)} =\frac{b}{sin(B)}[/tex]
substitute and solve for B
[tex]\frac{13}{sin(59\°)} =\frac{14}{sin(B)}\\ \\sin(B)=14*sin( 59\°)/13\\ \\sin(B)=0.9231\\ \\B=arcsin(0.9231)\\ \\B=67.4\°[/tex]
Remember that the angle B can take two values
[tex]B=180\°-67.4\°=112.6\°[/tex]
Step 2
Find the value of angle C
Remember that
the sum of the internal angles of a triangle is equal to [tex]180\°[/tex]
so
[tex]A+B+C=180\°[/tex]
we have
[tex]A=59\°[/tex]
[tex]B=112.6\°[/tex]
substitute and solve for C
[tex]59\°+112.6\°+C=180\°[/tex]
[tex]C=180\°-(59\°+112.6\°)=8.4\°[/tex]
Step 3
Find the measure of side c
Applying the law of sines
[tex]\frac{a}{sin(A)} =\frac{c}{sin(C)}[/tex]
substitute and solve for c
[tex]\frac{13}{sin(59\°)} =\frac{c}{sin(8.4\°)}[/tex]
[tex]c=\frac{13}{sin(59\°)}*sin(8.4\°)\\\\c=2.2\ units[/tex]
