Question

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

Answer 1

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

Answer 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