To the first question : It does not
The angle 5 pi/4 is in the third quadrant. In fact, it is a rotation away from the negative x axis by pi/4 radians.
The reference angle in the first quadrant is pi/4 . Since sin(pi/4) = sqrt(2)/2 and the sine of angles in the third quadrant are negative, it is seen that sin(5 pi/4) = - sqrt(2)/2. The same result can be easily read off of a unit circle chart.
The next step is to evaluate sin-1(- sqrt(2) /2 ) . Since sin( - pi/4) = - sqrt(2) /2 , we now have
sin-1( sin(-pi/4) ) and since - pi/4 satisfies he criterion listed below
sin-1( sin(-pi/4)) evaluates to simply - pi/4.
A useful criterion is that sin-1 (sin(x )) = x only works for
-pi/2 < x < pi/2
