Pre Cal

If the standard form is a + bi then the trigonometric form is r*(cos theta + i*sin theta)  where

 

r = sqrt(a^2 + b^2)  and  theta = arctan(b/a)

 

So, for 4 (squareroot) 3 - 4i  we have r = sqrt(16*3 + 16)  or r = 8

                                                    theta  = atan(-4/4*sqrt(3)) = -30°

                                                    

   

 4 (squareroot) 3 - 4i → 8(cos(30°) - i*sin(30°))  

So would that be 8(cos 11pi/6 + isin 11pi?6)

 

or 

 

8(cos 2pi/3 + isin 2pi/3)

z = 4 ( 3 - 4i)^1/2

Algebraic form:
z = 6-8i

Exponential form:
z = 10 × ei (-53°7'48″)

Trigonometric form:
z = 10 × (cos (-53°7'48″) + i sin (-53°7'48″))

Polar form:
r = |z| = 10
φ = arg z = -53.1301° = -53°7'48″ = -0.29517π

 

 

z = –12 – 12 ( 3i)^1/2

Algebraic form:
z = -12-18i

Exponential form:
z = 21.6333077 × ei (-123°41'24″)

Trigonometric form:
z = 21.6333077 × (cos (-123°41'24″) + i sin (-123°41'24″))

Polar form:
r = |z| = 21.63331
φ = arg z = -123.69007° = -123°41'24″ = -0.68717π