Use the Law of Sines to solve for all possible triangles that satisfy the given conditions.

Question

Use the Law of Sines to solve for all possible triangles that satisfy the given conditions. (If an answer does not exist, enter DNE. Round your answers to one decimal place. Below, enter your answers so that ∠B₁ is larger than ∠B₂.)

a=37 c=46 Angle A=33 degrees

find

Angle B1, Angle C1, b1

Angle B2, Angle C2, b2

Answer 1

Law of Sines

SinC/c = SinA/a = SinB/b

where c is the side opposite angle C

a is the side opposite angle A

b is the side opposite angle A

two possible triangles in this problem

in all similar problems there are either 0,1 or 2 possible triangles. if calculations show an angle's Sine greater than 1 or less than -1, there are zero triangles possible

ratio of Sines to Sides is a constant

SinC/c = SinA/a

SinC = cSinA/a

SinC = 46Sin33/37

SinC = (46/37)Sin33

C = about 42.62 degrees

or

C = 180-42.62= 137.38 degrees

sum of 3 angles = 180

B= 180-33-42.62= 180-75.62= 104.38 degrees

or

B = 180-33-137.38 = 180- 170.38 = 9.62 degrees

AB1C1 = 33, 104.4, 42.6

or

AB2C2 = 33, 9.6, 137.4

b1 = aSinB1/SinA = 37Sin104.4/Sin33

= about 65.8

b2 -aSinB2/SinA = 37Sin9.6/Sin33

= about 9.2

B1,C1,b1 = 104.4, 42.6, 65.8

B2,C2,b2 = 9.6, 137.4, 9.2

largest angle is opposite the largest side

smallest angle is opposite the smallest side

middle angle is opposite the middle sized side

if your solutions don't match that order, there's

a mistake somewhere

a diagram helps, but not necessary. the algebra, arithmetic or trigonometry

with the Law of Sines is sufficient, but a diagram is a good check on whether the answers make sense

(No guarantees the above calculations are not error free. the calculations are somewhat tedious, with room for errors, such as rounding errors every step of the way. But this is the general method.)