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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 sin^{0}*x* = cos^{0}*x* = 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 { }, which must eventually disappear

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

## HistoryThe formula was given by the French mathematician François Viète (1540 - 1603). |