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Trigonometry De Moivre's Theorem Extended Sine

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Cosine
Sine
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  1. Sine Proof: dms9.gif




Sine

Beginning with DeMoivre's formula:
dm1.gif [2.1]

We find an expression for the sine:
dms1.gif [1.1]

Expanding the sine and noting that i2=−1, we get:
dms2.gif [1.2]
Dividing by i, and grouping the sines:
dms3.gif [1.3]

Bringing the cosine to the end, in anticipation of an expansion, and taking cos x out of the power to make it n-2k-2 (and multiplying by cos x):
dms4.gif [1.4]

If n is even, let n=2p, so substituting in Equation 1.4 and making the cosine a square, with a view to using cos2x=1-sin2x
dms5.gif [1.5]
If n is odd, let n=2p-1, so the power of the cosine becomes 2p-2k-3, so to make it a factor of 2, we multiply by the cos x, and drop it from the equation:
dms6.gif [1.6]

To accommodate both cases, we can write the final cosine as cosq x, where q is 0 when n is odd, and 1 when n is even:
dms7.gif [1.7]

Writing (cos2 x)p-k-1 as 1−sin2 x)p-k-1 and writing the expansion:
dms8.gif [1.8]

For a given k of the first sum, the k's will remain unchanged throughout, so we can group the powers of (-1) and of the sines:
dms9.gif [1.9]
Where p is the ceiling (n/2), or pIsCeil.gif, and q=0, when n is odd, and q=1 when n is even.


A formula for the term with the power r of sine (that is, of sinrx) of sin nx is:
dms10.gif [1.10]
Where u=floor(r/2) and p=ceiling(n/2).





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