What are the values for this and how do I solve to get them?

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 ∠A₁ is smaller than ∠A₂.)

b = 45,    c = 44,    ∠C = 36°

∠A1 = ∠A_{2 =}

∠B1 = ∠B_{2 =}

a1 = a2 =

Answer 1

you posted this question more than once. I answered it using law of sines in the other post

but here's an alternative "estimate" or check on that answer that is surprisingly accurate

without a calculator, with trig, sine and inverse sine functions

you can estimate the answer to virtually the correct answer

44 and 45 are so close, that the opposite angles will also be similarly close

c=44, b=45, so if C=36, the B=about 37, just slightly more than 36

if one side is longer, then the opposite angle is also larger

but B also = 180-37 = 143. there's 2 triangles, one acute, the other obtuse

then A = 180-36-37 = 180-73 = 107 degrees, but also 180-143-36 = about 1 degree

that gives you all 3 of 2 triangles' angles, all you're missing is side a, opposite angle A

with the obtuse triangle, angles 1,36,143, side a would slightly more than 45-44 = 1+

maybe 1.1 or 1.2

that leaves just one side missing for a in the acute triangle. a has to be less than b+c or <44+45=99

45 < a < 99

about the middles is (45+99)/2 = 144/2 = 72 = a wild guess + or - 6 or so

to one decimal place a=71.6, using a calculator. turns out 72 was correct to the nearest integer

use the law of sines and a calculator to narrow it down more

a/sinA = b/sinB = c/sinC

a=BC, b=AC, c=BC

a = csinA/sinC = 44sin107/sin36

sin36 = between sin30 and sin45 = between about .7 and .5 or about .6

a=44sin107/.6

sin107 = sin(180-107)=sin73>sin60=about 1.732/2 =.866

sin73 = maybe about .9

a =44(.9)/.6=44(3/2) =22(3)=about 66. that's off by about 6

but it's a good check on whether your calculator answer is in the right range or not