Lesson 6.5: Trigonometric Form of a Complex Number
In this lesson, you will learn how to plot complex numbers in the complex plane and find absolute values of complex numbers, write trigonometric forms of complex numbers, multiply and divide complex numbers written in trigonometric form, use DeMoivre's Theorem to find powers of complex numbers, and find nth roots of complex numbers.
The Complex Plane
Just as real numbers can be represented by points on the real number line, you can represent a complex number z = a + bi as the point (a, b) in a coordinate plane (the complex plane). The horizontal axis is called the real axis and the vertical axis is called the imaginary axis. The absolute value of a complex number a + bi is defined as the distance between the origin (0,0) and the point (a,b).
Definition of the Absolute Value of a Complex Number
The absolute value of the complex number z = a + bi is given by |a + bi| = √(a² + b²). |
Trigonometric Form of a Complex Number
To work effectively with powers and roots of complex numbers, it is helpful to write complex numbers in trigonometric form. By letting θ be the angle from the positive real axis (measured counterclockwise) to the line segment connecting the origin and the point (a,b), you can write a = rcosθ and b = rsinθ where r = √(a² + b²). Consequently, you have a + bi = (rcosθ) + (rsinθ)i from which you can obtain the trigonometric form of a complex number. The trigonometric form of the complex number 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
Suppose you are given two complex numbers z₁ = r₁(cosθ₁ + isinθ₁) and z₂ = r₂(cosθ₂ + isinθ₂).
The product of z₁ and z₂ is z₁z₂ = r₁r₂[cos(θ₁ + θ₂) + isin(θ₁ + θ₂)].
The quotient of z₁ and z₂ is z₁/z₂ = r₁/r₂[cos(θ₁ - θ₂) + isin(θ₁ - θ₂)], z₂ does not equal 0.
Suppose you are given two complex numbers z₁ = r₁(cosθ₁ + isinθ₁) and z₂ = r₂(cosθ₂ + isinθ₂).
The product of z₁ and z₂ is z₁z₂ = r₁r₂[cos(θ₁ + θ₂) + isin(θ₁ + θ₂)].
The quotient of z₁ and z₂ is z₁/z₂ = r₁/r₂[cos(θ₁ - θ₂) + isin(θ₁ - θ₂)], z₂ does not equal 0.
nth Roots of a Complex Number
For a positive integer n, the complex number z = r(cosθ + isinθ) has exactly n distinct nth roots given by n√(r)(cos(θ + 2πk)/n + isin(θ + 2πk)/n where
k = 0, 1, 2, ...., n-1.
For a positive integer n, the complex number z = r(cosθ + isinθ) has exactly n distinct nth roots given by n√(r)(cos(θ + 2πk)/n + isin(θ + 2πk)/n where
k = 0, 1, 2, ...., n-1.