# The Trigonometric Form of Complex Numbers

## Key Questions

#### Explanation:

To convert a complex number

$z = x + i y$

to the polar form

$z = r \left(\cos \theta + i \sin \theta\right)$

Apply the following :

$\left\{\begin{matrix}r = | z | = \sqrt{{x}^{2} + {y}^{2}} \\ \cos \theta = \frac{x}{| z |} \\ \sin \theta = \frac{y}{| z |}\end{matrix}\right.$

And to convert

The polar form

$z = r \left(\cos \theta + i \sin \theta\right)$

to the standard form

$z = x + i y$

Apply the folowing

$\left\{\begin{matrix}x = r \cos \theta \\ y = r \sin \theta\end{matrix}\right.$

• The rectangular form of a complex form is given in terms of 2 real numbers a and b in the form: z=a+jb
The polar form of the same number is given in terms of a magnitude r (or length) and argument q (or angle) in the form: z=r|_q
You can "see" a complex number on a drawing in this way:

In this case the numbers a and b become the coordinates of a point representing the complex number in the special plane (Argand-Gauss) where on the x axis you plot the real part (the number a) and on the y axis the imaginary (the b number, associated with j).
In polar form you find the same point but using the magnitude r and argument q:

Now the relationship between rectangular and polar is found joining the 2 graphical representations and considering the triangle obtained:

The relationships then are:
1) Pitagora's Theorem (to link the length r with a and b):
$r = \sqrt{{a}^{2} + {b}^{2}}$
2) Inverse trigonometric functions (to link the angle q with a and b):
$q = \arctan \left(\frac{b}{a}\right)$

I suggest to try various complex numbers (in diferente quadrants) to see how these relationships work.

• Yes, of course.

Polar form is very convenient to multiply complex numbers.
Assume we have two complex numbers in polar form:
${z}_{1} = {r}_{1} \left[\cos \left({\phi}_{1}\right) + i \cdot \sin \left({\phi}_{1}\right)\right]$
${z}_{2} = {r}_{2} \left[\cos \left({\phi}_{2}\right) + i \cdot \sin \left({\phi}_{2}\right)\right]$
Then their product is
${z}_{1} \cdot {z}_{2} = {r}_{1} \left[\cos \left({\phi}_{1}\right) + i \cdot \sin \left({\phi}_{1}\right)\right] \cdot {r}_{2} \left[\cos \left({\phi}_{2}\right) + i \cdot \sin \left({\phi}_{2}\right)\right]$
Performing multiplication on the right, replacing ${i}^{2}$ with $- 1$ and using trigonometric formulas for cosine and sine of a sum of two angles, we obtain
${z}_{1} \cdot {z}_{2} = {r}_{1} {r}_{2} \left[\cos \left({\phi}_{1} + {\phi}_{2}\right) + i \cdot \sin \left({\phi}_{1} + {\phi}_{2}\right)\right]$
The above is a polar representation of a product of two complex numbers represented in polar form.

Raising to any real power is also very convenient in polar form as this operation is an extension of multiplication:
${\left\{r \left[\cos \left(\phi\right) + i \cdot \sin \left(\phi\right)\right]\right\}}^{t} = {r}^{t} \left[\cos \left(t \cdot \phi\right) + i \cdot \sin \left(t \cdot \phi\right)\right]$

Addition of complex numbers is much more convenient in canonical form $z = a + i \cdot b$. That's why, to add two complex numbers in polar form, we can convert polar to canonical, add and then convert the result back to polar form.
The first step (getting a sum in canonical form) results is
${z}_{1} + {z}_{2} = \left[{r}_{1} \cos \left({\phi}_{1}\right) + {r}_{2} \cos \left({\phi}_{2}\right)\right] + i \cdot \left[{r}_{1} \sin \left({\phi}_{1}\right) + {r}_{2} \sin \left({\phi}_{2}\right)\right]$

Converting this to a polar form can be performed according to general rule of obtaining modulus (absolute value) and argument (phase) of a complex number represented as $z = a + i \cdot b$ where
$a = {r}_{1} \cos \left({\phi}_{1}\right) + {r}_{2} \cos \left({\phi}_{2}\right)$ and
$b = {r}_{1} \sin \left({\phi}_{1}\right) + {r}_{2} \sin \left({\phi}_{2}\right)$

This general rule states that
$z = r \left[\cos \left(\phi\right) + i \cdot \sin \left(\phi\right)\right]$ where
$r = \sqrt{{a}^{2} + {b}^{2}}$ and
angle $\phi$ (usually, in radians) is defined by its trigonometric functions
$\sin \left(\phi\right) = \frac{b}{r}$,
$\cos \left(\phi\right) = \frac{a}{r}$
(it's not defined only if both $a = 0$ and $b = 0$).
Alternatively, we can use these equations to define angle $\phi$:
If $a \ne 0$, $\tan \left(\phi\right) = \frac{b}{a}$. Or, if $b \ne 0$, $\cot \left(\phi\right) = \frac{a}{b}$.

• Trigonometric Form of Complex Numbers

$z = r \left(\cos \theta + i \sin \theta\right)$,

where $r = | z |$ and $\theta =$Angle$\left(z\right)$.

I hope that this was helpful.