Almost every trig and trig-like problem involves the 30/60/90 or 45/45/90 triangle. This one is the latter.
1-i corresponds to the point (1,-1), right there in the fourth quadrant, magnitude sqrt{2}, angle -45^circ .
Trigonometric form is essentially polar form written rectangularly, as r( cos theta + i sin theta), often conveniently abbreviated r\ text{cis}\ theta .
1 = sqrt{2} (1/sqrt{2}) = sqrt{2}\ cos(- 45^circ)
-1 = sqrt{2} (-1/sqrt{2}) = sqrt{2}\ sin(-45^circ)
1 -i = sqrt{2}( cos(- 45^circ) + i \ sin(-45^circ) ) = sqrt{2}\ text{cis}(-45^circ)
Using our goofy system where the circle constant pi is half a circle, in radians that's just
1 -i = \sqrt{2} \ text{cis}(-pi/4)