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Sine of multiples

The general formula for calculating the sine of multiples is

 


Explanation

We start with

Let y = y + z then you get

For x = y = z it follows

There seems no clear structure in that. Because both sin0x = cos0x = 1 you can expand all terms by this factor

We try to find some kind of Pascal's triangle. Because of that, we add by brute force terms with brackets { } toe, that must finally disappear again

If we now also add a factor that is alternately 0, 1, 0, −1 and so on, it is solved. Therefor we can write Vieta's formula for calculating the sine of multiples as

 


History

The formula was given by the French mathematician François Viète (1540 - 1603).


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