Question

Two triangles can be formed with the given information. Use the Law of Sines to solve the triangles. A = 59°, a = 13, b = 14

Answer 1

First, you have to know what is the law of Sine? 
(Refer to this website for more information: https://www.mathsisfun.com/algebra/trig-sine-law.html )

a/sin A = b/sin B = c/sin C
in this case: 13/sin59=14/sinB, => sinB=(sin59)*14)/13 use your calculator: 
B is about 67 degree.
the three angles make a total of 180 degree, so C=180-59-67=54
again, use law of sin, to find out c
13/sin59=c/sin54 =>c is about 12

I don't see how to form the other triangle.

Answer 2

Answer:

First triangle

$B=67.4\°$

$C=53.6\°$

$c=12.2\ units$

Second triangle

$B=112.6\°$

$C=8.4\°$

$c=2.2\ units$

Step-by-step explanation:

In this problem we have

$A=59\°$

$a=13\ units$

$b=14\ units$

First Triangle

Step 1

Find the value of angle B

Applying the law of sines

$\frac{a}{sin(A)} =\frac{b}{sin(B)}$

substitute and solve for B

$\frac{13}{sin(59\°)} =\frac{14}{sin(B)}\\ \\sin(B)=14*sin( 59\°)/13\\ \\sin(B)=0.9231\\ \\B=arcsin(0.9231)\\ \\B=67.4\°$

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 $180\°$

so

$A+B+C=180\°$

we have

$A=59\°$

$B=67.4\°$

substitute and solve for C

$59\°+67.4\°+C=180\°$

$C=180\°-(59\°+67.4\°)=53.6\°$

Step 3

Find the measure of side c

Applying the law of sines

$\frac{a}{sin(A)} =\frac{c}{sin(C)}$

substitute and solve for c

$\frac{13}{sin(59\°)} =\frac{c}{sin(53.6\°)}$

$c=\frac{13}{sin(59\°)}*sin(53.6\°)\\\\c=12.2\ units$

Second Triangle

Step 1

Find the value of angle B

Applying the law of sines

$\frac{a}{sin(A)} =\frac{b}{sin(B)}$

substitute and solve for B

$\frac{13}{sin(59\°)} =\frac{14}{sin(B)}\\ \\sin(B)=14*sin( 59\°)/13\\ \\sin(B)=0.9231\\ \\B=arcsin(0.9231)\\ \\B=67.4\°$

Remember that the angle B can take two values

$B=180\°-67.4\°=112.6\°$

Step 2

Find the value of angle C

Remember that

the sum of the internal angles of a triangle is equal to $180\°$

so

$A+B+C=180\°$

we have

$A=59\°$

$B=112.6\°$

substitute and solve for C

$59\°+112.6\°+C=180\°$

$C=180\°-(59\°+112.6\°)=8.4\°$

Step 3

Find the measure of side c

Applying the law of sines

$\frac{a}{sin(A)} =\frac{c}{sin(C)}$

substitute and solve for c

$\frac{13}{sin(59\°)} =\frac{c}{sin(8.4\°)}$

$c=\frac{13}{sin(59\°)}*sin(8.4\°)\\\\c=2.2\ units$