Step 
Arcsine  Arccosine  Arctangent 
Locate the input variable value on the appropriate axis in the plane of the unit circle:  Use $(0,y)$ on the vertical axis of the unit circle plane 
Use $(x,0)$ on the horizontal axis of the unit circle plane.  Use $(1,y)$ on the axis $x =1$ in the unit circle plane. 
Reverse the appropriate projection to find a unique point $P(t)$ on the unit circle. 
Project parallel to the horizontal axis to a point $P(t) = (a(y),b(y))$ on the unit circle with $a(y) \ge 0$. 
Project parallel to the vertical axis to a point $P(t) = (a(x),b(x))$ on the unit circle with $b(x) \ge 0$.  Project toward the point $O =(0,0)$ to a point $P(t) = (a(y),b(y))$ on the unit circle with $a(y) \gt 0$. 
Find the measure $t$ of the arc on the unit circle in the appropriate interval so that $P(t) = (a,b)$ 
The measure $t$ of the arc on the unit circle is the arcsine of $y$,: $t = \arcsin(y)$ 
The measure $t$ of the arc on the unit circle is the arccosine of $x$,: $t = \arccos(x)$  The measure $t$ of the arc on the unit circle is the arctangent of $y$: $t = \arctan(y)$ 
Unit Circle Diagram 
arcsine

arccosine

arctangent
