6.5 Trigonometric Form Of A Complex Number
The absolute value of a complex number a + bi is defined as the distance between the origin (0, 0) and the point (a, b)∣a + bi∣ = √(a² + b²)
for a real number it is also true ∣±a∣= a
for a real number it is also true ∣±a∣= a
Trigonometric Form of a Complex Number
The trigonometric form of z= a + bi is given by
z = r(cosθ + isinθ)
where a = rcosθ, b = rsinθ, r = √(a² + b²), and tanθ = b/a. The number r is the modulus of z, and θ is called an argument of z.
The trigonometric form of z= a + bi is given by
z = r(cosθ + isinθ)
where a = rcosθ, b = rsinθ, r = √(a² + b²), and tanθ = b/a. The number r is the modulus of z, and θ is called an argument of z.
Multiplication and Division of Complex Numbers
Powers of Complex Numbers
The trigonometric form of a complex number is used to raise a complex number to a power.
DeMoivre's Theorem
If z = r(cosθ + isinθ) is a complex number and n is a positive integer, then
zⁿ = [r(cosθ + isinθ)]ⁿ
= rⁿ(cosnθ + isinnθ)
The trigonometric form of a complex number is used to raise a complex number to a power.
DeMoivre's Theorem
If z = r(cosθ + isinθ) is a complex number and n is a positive integer, then
zⁿ = [r(cosθ + isinθ)]ⁿ
= rⁿ(cosnθ + isinnθ)
Roots of Complex Numbers
For positive integer, the complex number z =r(cosθ +isinθ) has exactly n distinct nth roots given byr^1/n(cos(θ+2πk)/n isin(θ+2πk)/n)
For positive integer, the complex number z =r(cosθ +isinθ) has exactly n distinct nth roots given byr^1/n(cos(θ+2πk)/n isin(θ+2πk)/n)