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Sine

The sine can be expressed with exponential functions as

 

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Explanation

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 braces are two power series for exponential functions, and thus

so that

 

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Geometric explanation

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

Therefore, for the sine applies

i · sin x = ½ (e^ix − e^−ix)

 

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Example 1

You can see that sin (2π) = 0, as

 

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Example 2

You can see that sin (½π) = 1, as

 

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Example 3

You can see that sin² (¾π) = 0,5, as

 

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History This formula for the sine was described by the Swiss mathematician Leonhard Euler (1707 - 1783) .

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