how do i evaluate the function csc 11 pie/4 without using a calculator

Answer 1

Subtract 2 pi from (11/4) pi and you get (3/4)pi. That is in the second quadrant, where the sin and csc functions are positive.
The reference angle is
pi - (3/4)pi = pi/4
Thus we have
csc(11/4)pi = csc(3/4)pi = csc(pi/4)
= 1/sin(pi/4) = sqrt2
The sine is 1/sqrt2 because the reference angle is 45 degrees. You can get sine 45 from the Pythagorean theorem.