Question

How do you find a trigonometric form of a complex number?

Answer 1

Trigonometric Form of a Complex Number

The trigonometric form of a complex number z = a + bi is

z = r(cos θ + isin θ),

where r = |a + bi| is the modulus of z, and tan θ =

b

a

. θ is called the argument of z. Normally,

we will require 0 ≤ θ < 2π.

Examples

1. Write the following complex numbers in trigonometric form:

(a) −4 + 4i

To write the number in trigonometric form, we need r and θ.

r =

√

16 + 16 = √

32 = 4√

2

tan θ =

4

−4

= −1

θ =

3π

4

,

since we need an angle in quadrant II (we can see this by graphing the complex number).

Answer 2

{\displaystyle z=a+bi=r\left(\cos \phi \ +i\sin \phi \right)}

where

i is the imaginary number {\displaystyle \left(i\ ={\sqrt {-1}}\right)}

the modulus {\displaystyle r=\operatorname {mod} (z)=|z|={\sqrt {a^{2}+b^{2}}}}

the argument {\displaystyle \phi =\arg(z)} is the angle formed by the complex number on a polar graph with one real axis and one imaginary axis. This can be found using the right angle trigonometry for the trigonometric functions.

This is sometimes abbreviated as {\displaystyle r\left(\cos \phi \ +i\sin \phi \right)=r\operatorname {cis} \phi } and it is also the case that {\displaystyle r\operatorname {cis} \phi =re^{i\phi }} (provided that {\displaystyle \phi }is in radians). The latter identity is called Euler's formula.

Euler's formula can be used to prove DeMoivre's formula: {\displaystyle (\cos \phi \ +i\sin \phi )^{n}=\cos(n\phi )+i\sin(n\phi ).}This formula is valid for all values of n, real or complex.