PreCalc Complex Numbers vocab

DeMoivre;s THeorem states that [r(cosθ+i sinθ)]ⁿ =

rⁿ[cos(nθ)+i sin(nθ)]

r₁(cosθ₁+i sinθ₁)/r₂(cosθ₂+i sinθ₂)=

r₁/r₂[cos(θ₁-θ₂) +i sin(θ₁-θ₂)

The value √a²+b² is the __________ of the complex number a+bi

absolute value

The quotient of two complex numbers in polar form is found by _____________ their moduli and _____________ their arguments

dividing, subtracting

In the polar form of the complex number r(cosθ+ i sinθ), r is called the __________ and θ is called the ______________, 0≤θ≤2π

modulus, argument

The product of two complex numbers in polar form is found by _______ their moduli and _________ their arguments

multiplying, adding

In a complex plane, the horizontal axis is referred to as the ________ and the vertical axis is referred to as the _______ axis.

real, imaginary

r₁(cosθ₁+i sinθ₁) * r₂(cosθ₂+i sinθ₂)=

r₁r₂[cos(θ₁+θ₂) +i sin(θ₁+θ₂)]

polar form of a complex number

w=r(cosθ+ i sinθ) r=√a²+b² tanθ= b/a a=r cosθ b=r sinθ

pattern that i found in some of the roots answers

z₀= a+bi z₁=-b+ai z₂=-a-bi z₃=b-ai

complex cube roots of 1

z₀=1 z₁=-.5+.9i z₂=-.5-.9i

complex cube roots of -i

z₀=i z₁=-.9-.5i z₂=.9-.5i

absolute value of complex number (z)

|a+bi|= √a²+b²

DeMoivre's THeorem for finding Complex Roots (Theorem on nth Roots)

ⁿ√r[cos((θ+2πk)/n)+i sin((θ+2πk)/n)] k=0,1,2,3

To convert a complex number from rectangular form, z=a+bi, to polar form, z=r(cosθ+i sinθ), we use the relationships r=___________ and tanθ=___________ noting the quadrant in which the graph of z lies.

√a²+b² , b/a