Section 5.3 – The Complex Plane; De Moivre’s Theorem

Real Axis Imaginary Axis Ox y Complex Plane

Cartesian Form Polar Form

Plot 3 – 4i in the complex plane and write it in polar form. Express the argument in degrees.

z = 3 – 4i -4 3 z = x +yi θ z = rcosθ +(rsinθ)i α z = r(cosθ +isinθ) z = 5(cos o +isin o ) imaginary axis real axis

Write the point in rectangular form r = 3 θ = 3π/2 z = x + yi x = r cosθy = r sinθ x = 3 cos(3π/2) x = 3(0) = 0 y = 3 sin(3π/2) y = 3(-1) = 3 z = 0 - 3i z = - 3i

Theorem

If z = 3(cos130 o + isin130 o ) and w = 4(cos270 o + isin270 o ), what is zw? r1= 3, r2= 4 θ1 = 130 o, θ2 = 270 o zw = r1r2[cos(θ1 + θ2) + isin(θ1 + θ2)] zw = (3)(4)[cos(130 o o ) + isin(130 o o )] zw = 12[cos(400 o ) + isin(400 o )] zw = 12[cos(40 o ) + isin(40 o )] 40 o and 400 o are coterminal

Theorem DeMoivre’s Theorem

Write the expression in standard form a + bi = z 6

r = 27, θ = 60 o

Theorem Finding Complex Roots

Find the complex fourth roots of θ α

Change 2π to 360 o since our angle is in degrees, set n = 4 since we are finding the complex fourth roots, plug in r = 2 and θ = 120 o

We can simplify the fractions

Use values of k from 0 to n-1, that is 0 to 3