A combination of both the real and imaginary numbers is called as complex numbers. It can be expressed in the mathematical form as 'a + bi', where a and b are the real numbers, and i is the imaginary unit number which is equal to √(-1). In this tutorial, you will learn how to convert and write complex numbers into trigonometric form with an example.

Let us consider a complex number ** -4 - 1i**.
Rewrite -1i as -i and hence it is -4 - i.
The trigonometric form of a complex number z can be written as |z|(cos θ + isin θ)
Where, |z| is the modulus and θ is the angle created on the complex plane.
The modulus of a complex number is the distance from the origin on the complex plane.
|z| = √(a^{2} + b^{2})
Where,
z = a + bi

** Step 1 : **
Find |z| :
Substituting the actual values of a = -4 and b = -1 in the modulus formula, we get
|z| = √((-1)^{2} + (-4)^{2})

** Step 2 : **
|z| = √(1 + (-4)^{2})
|z| = √ (1+16)

** Step 3 : **
Adding 1 and 16 we get 17. Hence,
|z| = √17

**Step 4 : **
Find θ:
The angle of the point on the complex plane is the inverse tangent of the complex portion over the real portion.

θ = 57.2957795 * arctan(-1 / -4)

Since inverse tangent of ((-1) / (-4)) produces an angle in the third quadrant, the value of the angle is 194.0362. θ = 194.0362

**Step 5 : **
Substitute the values of θ = 194.0362 and |z| = √17 to write -4 - i as |z|(cos θ + isin θ).

**|z| = √17 (cos (194.0362) + isin (194.0362))** is the trigonometric form of the complex number.