Complex Numbers in Trigonometric Form

Step right into this walkthrough where we'll peel back the layers of complexity surrounding complex numbers. Our mission today? To guide you through the transformation of a complex number from its algebraic to its trigonometric form (or polar form).

Imagine a complex number in trigonometric form. It looks something like this: $$ z = r \cdot ( \cos \alpha + i \cdot \sin \alpha ) $$ where 'r' is the magnitude of the complex number and 'alpha' is its argument or phase.

Here's a tip for you to keep in mind. When you're dealing with complex numbers, like z=a+bi, they can be viewed as points on the Gaussian plane, z=(a;b).

Think of the magnitude ( or modulus ) of a complex number z=a+bi as the length of an imaginary vector. This vector shoots straight from the origin to the point that corresponds to your complex number.

Now, to calculate this, there's no need to reinvent the wheel. The trusty old Pythagorean theorem is up for the task$$ r = \sqrt{a^2 + b^2} $$

The argument of your complex number, meanwhile, is nothing but the angle nestled between the positive real axis and the vector you're working with. Whip out the arctangent of b/a to get your hands on this: $$ θ = \arctan(b/a) $$

What's neat is that complex numbers, when expressed in trigonometric (aka polar) form, become a piece of cake to manipulate, streamlining many complex number operations. Plus, it's this form that gives you the luxury to represent complex numbers in an exponential fashion. All in all, a pretty cool and practical way to look at things, don't you think?

Example

So, How Does This Work in Practice?

Let's roll up our sleeves and delve into a real-world example.

We'll start by sketching the complex number z1=4+3i on the Gauss plane.

You'll spot our complex number sitting pretty at the coordinates (4,3) on the plane.

Let's now draw a vector "v" that links the origin (O) to our point (4,3) on the plane.

The length of this vector? That's the magnitude of our complex number.

The angle 'alpha', the one snuggled between the vector and the positive real semi-axis, that's the argument of our complex number.

Every complex number plotted on the plane is blessed with a magnitude 'r' and an argument 'α'.

Together, these two values [r;α] provide the polar coordinates of the complex number.

From Algebraic to Trigonometric Form

Switching from the algebraic form of the complex number z=a+bi to the trigonometric form requires you to find two key things: the magnitude's length and the angle 'alpha' of the argument.

Fortunately, we have Pythagoras to thank for a theorem that makes calculating the magnitude a breeze.

Take, for example, the complex number z₁=4+3i. It forms a triangle on the plane.

The magnitude represents the hypotenuse of the triangle, while the other two sides, with lengths a=4 and b=3, form the legs.

Thanks to Pythagoras, the hypotenuse of the triangle works out as follows:

$$ r = \sqrt{a^2+b^2} = \sqrt{4^2+3^2} = \sqrt{16+9} = \sqrt{25} = 5 $$

Which means that our complex number has a magnitude of r=5.

Now, About the Argument.

To uncover the angle 'alpha' (the argument), we'll need to dip our toes into a little trigonometry.

It can be found using the arctangent of the ratio b/a, which in this case is 3/4.

$$ \alpha = \arctan ( \frac{b}{a} ) $$

$$ \alpha = \arctan ( \frac{3}{4} ) = 36.87° $$

This tells us that the argument of the complex number is a=36.87°.

Putting It All Together. Marrying these two pieces of information, we discover the polar coordinates of the complex number z₁

$$ z_1 = [ \ r \ ; \ \alpha \ ] = [ \ 5 \ ; \ 36.87° \ ] $$

With the magnitude r=5 and the argument α=36.87° now in our grasp, we can elegantly rewrite the complex number in its trigonometric form:

$$ z_1 = r \cdot ( \cos \alpha + i \cdot \sin \alpha ) $$

$$ z_1 = 5 \cdot ( \cos 36.87° + i \cdot \sin 36.87° ) $$

In other words, stating z₁=4+3i is no different from saying z₁=5·[cos 36.87+i·sin 36.87°].

They're simply two ways of expressing the same complex number.

From Trigonometric to Algebraic Form

Suppose you're faced with a complex number in trigonometric form:

$$ z = 5 \cdot ( \cos \alpha + i \cdot \sin \alpha ) $$

To journey back to its algebraic form z=a+bi, you can make use of these two handy formulas:

$$ a= r \cdot \cos \alpha $$

$$ b= r \cdot \sin \alpha $$

For instance, if the magnitude is r=5 and the argument is α=36.87°

$$ a= 5 \cdot \cos 36.87° = 4$$

$$ b= 5 \cdot \sin 36.87° = 3 $$

With this, we find our complex number in its algebraic form at coordinates (a;b)=(3;4), or z₁=3+4i.

Why should we bother with the polar form of complex numbers?

Complex numbers in their polar form are a game changer in a variety of applications.

This has to do with the fact that many operations involving complex numbers are made substantially simpler when we express these numbers in polar form rather than sticking with their Cartesian form.

Imagine, if you will, a pair of complex numbers expressed in polar form.

$$ z_1 = r_1(\cos(θ_1) + i \cdot \sin(θ_1)) $$

$$ z_2 = r_2(\cos(θ_2) + i \cdot \sin(θ_2)) $$

If you wish to multiply these two complex numbers, it's as straightforward as multiplying their magnitudes and adding their angles together.

$$ z_1 \cdot z_2 = r_1 \cdot r_2 ( \cos(θ_1+θ_2) + i \cdot \sin(θ_1+θ_2)) $$

Likewise, division becomes a breeze: you merely divide the magnitudes and calculate the difference between their angles.

$$ \frac{z_1}{z_2} = \frac{r_1}{r_2} ( \cos(θ_1-θ_2) + i \cdot \sin(θ_1-θ_2)) $$

Moreover, the polar form is your golden ticket to the world of the exponential form of complex numbers, courtesy of Euler's formula.

Remember, if you found this session insightful, Nigiara has plenty more online lessons to keep you engaged and informed. Keep checking back for more!