I thinhk you are totally confused and have seriously stuffed up this question.
I will discuss it though.
arc sin cannot be greater than 1 so I will change the expression.......
sin(arcsin(4/5)) there is only one answer, 4/5
Lets look at your second part and see if that makes more sense
sin(pi/6) = 1/2 (not sqrt3/2) and cos(pi/6) =sqrt3/2 (not 1/2)
I just draw the right angled triangle and also when necessary think about the sign
You don't have a terminal point for the sin of an angle
You have a terminal point on the unit circle for an angle.
the terminal point for pi/6 is ((sqrt3)/2 , 0.5)