Write the complex number in trigonometric form? using both degrees and radians 1 − i

1 Answer
May 11, 2018

1 -i

= sqrt{2}( cos(- 45^circ) + i \ sin(-45^circ) )

= sqrt{2}\ text{cis}(-45^circ)

= \sqrt{2} \ text{cis}(-pi/4)

Explanation:

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)