Question

Find the area of a non right triangle with the side lengths 1 and square root (2/3) and a non included angle of 45 degrees.

Answer 1

I see two possible cases for the triangle. In both cases, since
the angle given is not contained between the two given sides, you would
have the ambiguous case, that is, there may be two triangles possible for
each interpretation.

Case 1:
sketch a triangle ABC, so that angle B = 45° , and AC, the side opposite angle B, is √(2/3) , and AB = 1
Let's find angle C
sinC/1 = sin45/√(2/3)
sinC = .866025... (looks familiar)
angle C = 60° or angle C = 120°

If angle C = 60, then angle A = 75° and
area = (1/2)(1)(√(2/3) sin75
= .3943 square units

if angle C = 120, then angle A = 15° and
area = (1/2)(√(2/3)sin15°
= appr .10566...

Case 2:
I sketched triangle ABC, made angle B = 45, but this time
let AC = 1 and BC = √(2/3)

Follow the same steps I used in Case 1