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The sine can be expressed with exponential functions as



The power series for the sine has only odd exponents

We multiply all terms with 2i and get

By conversion with the imaginary unit all terms obtain a plus sign

You can also write it as

We add even exponents and subtract these immediately

Rearranging gives

In brackets are two power series for exponential functions, and thus

so that


Geometric explanation

The unit circle allows you to calculate the vertical distance between the points eix and e−ix as eix − e−ix |.

Therefore, for the sine applies

i · sin x = ½ (eix − e−ix)


Example 1

You can see that sin (2π) = 0, as


Example 2

You can see that sin (½π) = 1, as


Additional information

This formula for the sine was described by the Swiss mathematician Leonhard Euler.


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