Verified answer

Dismantling from the inside out.

There are 2pi radians in a complete circle (or 360 degrees). That's the same as 8pi/4. What's left after taking out one full revolution is 11-8 = 3. So the working angle is 3pi/4. Pi/4 is 45 degrees. So we are looking at 3 times 45 = 135 degrees. Now to find the cotangent, we need to think cos/sin. For 45 degrees, the numerical values of sin and cos are the same - the sign is the only issue (both have absolute values of sqrt(2)/2 = 0.707...).

135 degrees is in the second quadrant. The sin is still positive (positive y or vertical), but the cos is negative (negative x or horiz). Since a negative divided by a positive is always negative, we would have -0.707~/0.707~ = -1 (recalling that the absolute values, while irrational, are still identical, and any number divided by itself is 1.)

Now we are asked to find the angle whose sin is this value, that is, -1. Sin reaches a value of -1 at 270 degrees (or 3pi/2). Any other angle that differs from 3pi/2 by exactly 2pi would also solve this expression. But we usually restrict the answer for arcsin to 0 to 2 pi. So the right answer is 3pi/2.

Walk through the explanation, and make sure it makes sense. Probably the trickiest part is to recognize that 11pi/4 is actually a full rotation plus 3/8 more. Ergo 135 degrees.

cot(11pi/4) = cot((11pi/4) - 2pi) = cot(3pi/4) = -1.

So sin^-1(cot(11pi/4)) = sin^-1(-1) = -pi/2.

(Recall that the range of sin^-1 is [-pi/2, pi/2].)