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Cosine

The cosine can be expressed with complex exponential functions as

 

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Explanation

In the power series for the complex cosine are only even exponents

so we can use the imaginary unit to give all terms a plus sign

After duplicating all terms you get

To this we add odd exponents and subtract these immediately

Reshuffling gives

In braces are two power series for exponential functions and thus

so that

 

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

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