Euler's Formula

Euler’s formula establishes the fundamental relationship between the trigonometric functions and the complex exponential function. Euler’s formula or Euler’s equation is one of the most fundamental equations in maths and engineering and has a wide range of applications.

Euler’s Formula Equation

Euler’s formula or Euler’s identity states that for any real number x,

eix = cos x + i sin x


  • x = real number
  • e = base of natural logarithm
  • sin x & cos x = trigonometric functions
  • i = imaginary unit

Note: The expression cos x + i sin x is often referred to as cis x.

Example Question Using Euler’s Equation Formula

Question: Find the value of e i π/2.


Given ei π/2

Using Euler’s formula,

eix = cos x + i sin x

e i π/2 = cos π/2 + i sin π/2

e i π/2 = 0 + i × 1

e i π/2 = i

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