The two sides of a triangle are 2 and 3, and the sine of the obtuse angle between these sides is 2√2 / 3.

The two sides of a triangle are 2 and 3, and the sine of the obtuse angle between these sides is 2√2 / 3. Find the third side of the triangle.

Let’s call the triangle ABC, where AB = 3, AC = 2 and sinA = 2√2 / 3
Find cosA ^
cosA = √ (1 – sin ^ 2 A) = √ (1 – 4 * 2/9) = √ (1 – 8/9) = √1 / 9 = 1/3
Now, by the cosine theorem, we find the third side of the BC:
BC ^ 2 = AB ^ 2 + AC ^ 2 – 2 * AB * AC * cosA
ВС ^ 2 = 3 ^ 2 + 2 ^ 2 – 2 * 3 * 2 * 1/3
BC ^ 2 = 9 + 4 – 12 * 1/3
BC ^ 2 = 13-4
BC ^ 2 = 9
BC = ± 3
-3 extraneous root
It turns out that the third party (BC) = 3
Answer: 3