First, recall the basic conversion formulas to convert from a complex number in a+bi format into trigonometric r*cis theta format. These come mainly from the rectangular to polar conversion methodology:
r = sqrt(a^2 + b^2)
tan theta_{ref} = |b/a| color(white)("aaaaaa")"Reference Angle"
In this problem, a = 1 and b = -1. We can find r readily:
r = sqrt(a^2 + b^2) = sqrt(1^2 + (-1)^2) = sqrt(1 + 1) = sqrt(2)
To find the proper theta angle, we begin with finding the reference angle theta_{ref} first:
tan theta_{ref} = |(-1)/(1)|
tan theta_{ref} = 1
theta_{ref} = pi/4
(This could have been readily apparent by noticing that both a and b are the same absolute value (1).)
If we consider that the complex number 1-i is graphed in Quadrant IV of the complex plane, and we recall how to use reference angles, we can see that the proper theta in this case is:
theta = 2pi - pi/4 = (7pi)/4
Thus, the complex number 1-i in trigonometric form is sqrt(2)*cis (7pi)/4, also written as sqrt(2)(cos ((7pi)/4) + i*sin ((7pi)/4))