Question

what are the angles on the unit circle for the function tan x = -1?

Answer 1

well 45 degrees would be +1 so we need to find quadrants where the horizontal and vertical components (x and y) have opposite signs right in the middle of the quadrant.

That would be in quadrant 2 (x is - and y is +) and quadrant 4 (x is + and y is -)
so try 90+45 = 135 (pi/2+pi/4 = 3 pi/4)
and
360 - 45 = 315 (2 pi - pi/4 = 7 pi/4)