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

An explanation would be gravely appreciated.

Answer 1

First triangle

`B=67.4\°`

`C=53.6\°`

`c=12.2\ units`

Second triangle

`B=112.6\°`

`C=8.4\°`

`c=2.2\ units`

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

`(a)/(sin(A)) =(b)/(sin(B))`

substitute and solve for B

`(13)/(sin(59\°)) =(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

`(a)/(sin(A)) =(c)/(sin(C))`

substitute and solve for c

`(13)/(sin(59\°)) =(c)/(sin(53.6\°))`

`c=(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

`(a)/(sin(A)) =(b)/(sin(B))`

substitute and solve for B

`(13)/(sin(59\°)) =(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

`(a)/(sin(A)) =(c)/(sin(C))`

substitute and solve for c

`(13)/(sin(59\°)) =(c)/(sin(8.4\°))`

`c=(13)/(sin(59\°))*sin(8.4\°)\\\\c=2.2\ units`