How do you find a trigonometric form of a complex number?
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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).
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{\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.
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