Before we start, let's remember the following
ASTC
This means that ALL (sin, cos and tan) are positive in the 1st quadrant
Sin is positive in the 2nd quadrant
Tan is positive in the 3rd quadrant
Cos is positive in the 4th quadrant
1) arcsin (-√3/2)
What angle has a sin of -√3/2?
From ASTC, above, Sin is negative in the 3rd and 4th quadrants
The angle is either 4π/3 or 5π/3.
2) arctan √3
What angle has a tan of √3
Tan is positive in the 1st and 3rd quadrants
The angle is either π/3 or 4π/3.
3) arcsin 0
What angle has a sin of 0
Sin is 0 at an angle of 0 or π
4) sin(arccos(-√3/2))
What is the sin of the angle that has a cos of -√3/2
Cos is negative in the 2nd and 3rd quadrants
The angle is 5π/6 or 7π/6
Sin(5π/6) = 1/2
Sin(7π/6) = -1/2
Refer to the unit circle for more on this.