Find a value for b such that the triangle has a)one solution b)two solutions c)no solution A=60 a=10

Answer 1

(a)

The triangle side is  and angle is .

The laws of sines : 

The angle is acute angle.

Relation from the ambiguous case:  , where  is altitude of the triangle.

If triangle has one solution and  is an acute angle , then  and .

 and .

Substitute ,  in .

If triangle has one solution, then values of  are  and .

Answer 2

(b)

The triangle side is  and angle is .

Relation from the ambiguous case:  .

If triangle has two solutions and  is an acute angle then .

Substitute the corresponding value in above condition.

.

Divide each side by .

Consider .

Here .

Thus, .

From (1) and (2), conclude that  .

If triangle has two solutions and  is an acute angle then .

Answer 3

(c)

The triangle side is  and angle is .

Relation from the ambiguous case: .

If triangle has no solution and  is an acute angle, then .

.

Divide each side by .

Substitute and in above expression.

.

If triangle has no solution then value of  is .