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

1 Answer
Dean R.
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)#

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