Trigonometric Form of a Complex Number - Tutorial, Example

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.

How to Write Complex Numbers in Trigonometric Form?

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| = √(a2 + b2) 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.

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