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Euler’s Formula

Euler’s formula establishes the fundamental relationship between the trigonometric functions and the complex exponential function. Euler’s formula states that, for any real number x:

The Euler’s formula for complex analysis states that :

\[\large e^{ix}=cos\;x+i\;sin\;x\]

Where,
$x$ =real number
$e$ =base of natural logarithm
$sin\;x$ & $cos\;x$ = trigonometric functions
$i$ =imaginary unit

Solved Examples

Question: Find the value of $e^{i\frac{\pi}{2}}$.

Solution:

Given $e^{i\frac{\pi}{2}}$

Using Euler’s formula,

$e^{ix}$ = $cos x$ + i $sin x$

$e^{i\frac{\pi}{2}}$ = $cos$ $\frac{\pi}{2}$ + i $sin$ $\frac{\pi}{2}$

$e^{i\frac{\pi}{2}}$ = 0 + i $\times$ 1

$e^{i\frac{\pi}{2}}$ = i