If cot x = 2 / 3 and x is in quadrant 4, find:

sin (x / 2)

Cos x / sin x = 2 / 3

2 sin x = 3 cos x

sin x = 3 cos x / 2

sin² x + cos² x = 1

9 cos² x / 4 + cos² x = 1 / * 4

9 cos² x + 4 cos² x = 4

13 cos² x = 4

cos x = 2 / √13

sin (x/2) = √ [ (1 - cos x) / 2] = √ [ (1 - 2/√13) / 2] = √ (√13 - 2) / (2√13)

or sin (x/2) ≈ 0.471858

Cot = adjacent / opposite = 2/3 but it lies in the 4th quadrant so 2/-3

Sin = opposite / hypotenuse, you have it as x/2

So you have the hypotenuse and the opposite is - 3 so - 3/2

Sin (-3/2) = - sin (3/2)

Alternative form : - 0.997495