Euler’s formula is a mathematical formula in complex analysis that establishes the fundamental relationship between the trigonometric functions and the complex exponential function.
Euler’s formula states that for any real number \(\theta\),
\(\boxed{e^{\displaystyle j\theta} = \cos(\theta) + j\sin(\theta)}\)
\(\boxed{e^{\displaystyle -j\theta} = \cos(\theta) - j\sin(\theta)}\)
where \(e\) is the Euler’s number (the base of natural logarithms) and \(j\) is the imaginary unit.
When \(\theta = \pi\), Euler’s formula may be rewritten as \(e^{\displaystyle\,j\pi} = -1\), which is known as Euler’s identity.