Question

Express the complex number in trigonometric form.

2 - 2i

Answer 1

Trigonometric form of the complex number : z = 2 - 2 i :
z = r ( cos t + i sin t )
r = | z | = √ ( 2² + (-2)² ) = √(4+4) = √8 = 2√2
t ( theta ) = tan^(-1) (-2 / 2 )= tan^(-1) (- 1 ) = - π / 4
z = 2√2 · ( cos (-π/4) + i sin (-π/4) ) =
= 2√2 · ( cos π/4 - i sin π/4 )

Answer 2

a+ib=2sqrt2(cos7pi/4+isin7pi/4)

Explanation:

a+ib=r (cos theta+isin theta)

r=sqrt a^2+b^2

r=sqrt (2)^2 +(-2)^2

r=2sqrt 2

theta=tan^-1(y/x) or (a/b)

theta= tan^-1(-2/2)

theta=-45 degrees

Now, I know that theta is in the fourth quadrant because cos (x-value) is positive. So, I am going to subtract my value from 360 degrees.

360-45= 315

theta=315

I can convert degrees to radians (if need be): 315 times pi/180= 7pi/4

Theta=7pi/4 r=2sqrt2

Substitute: a+ib=2sqrt2(cos7pi/4+isin7pi/4) this is radian format

or... a+ib=2sqrt2(cos 315+isin315) this is degree format