Question

Determine the number of possible solutions for a triangle with B=37 degrees, a=32, b=27

Answer 1

The guy above is right, 2 solutions.

Explanation:

In order to know how many solutions are possible in a triangle, we need to compare the b and height.

height in this question is: sin B * a

sin 37 * 32 = 19.2

In this case, The length of the line b is longer than the height,

which can create 2 triangles.

Answer 2

Two possible solutions

Explanation:

we know that

Applying the law of sines

`(a)/(sin(A))=(b)/(Sin(B))=(c)/(Sin(C))`

we have

`a=32\ units`

`b=27\ units`

`B=37\°`

step 1

Find the measure of angle A

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

substitute the values

`(32)/(sin(A))=(27)/(Sin(37\°))`

`sin(A)=(32)Sin(37\°)/27=0.71326`

`A=arcsin(0.71326)=45.5\°`

The measure of angle A could have two measures

the first measure------->
`A=45.5\°`

the second measure ----->
`A=180\°-45.5\°=134.5\°`

step 2

Find the first measure of angle C

Remember that the sum of the internal angles of a triangle must be equal to
`180\°`

`A+B+C=180\°`

substitute the values

`A=45.5\°`

`B=37\°`

`45.5\°+37\°+C=180\°`

`C=180\°-(45.5\°+37\°)=97.5\°`

step 3

Find the first length of side c

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

substitute the values

`(32)/(sin(37\°))=(c)/(Sin(97.5\°))`

`c=Sin(97.5\°)(32)/(sin(37\°))=52.7\ units`

therefore

the measures for the first solution of the triangle are

`A=45.5\°` ,
`a=32\ units`

`B=37\°` ,
`b=27\ units`

`C=97.5\°` ,
`b=52.7\ units`

step 4

Find the second measure of angle C with the second measure of angle A

Remember that the sum of the internal angles of a triangle must be equal to
`180\°`

`A+B+C=180\°`

substitute the values

`A=134.5\°`

`B=37\°`

`134.5\°+37\°+C=180\°`

`C=180\°-(134.5\°+37\°)=8.5\°`

step 5

Find the second length of side c

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

substitute the values

`(32)/(sin(37\°))=(c)/(Sin(8.5\°))`

`c=Sin(8.5\°)(32)/(sin(37\°))=7.9\ units`

therefore

the measures for the second solution of the triangle are

`A=45.5\°` ,
`a=32\ units`

`B=37\°` ,
`b=27\ units`

`C=8.5\°` ,
`b=7.9\ units`