  # Ken Ward's Mathematics Pages

## Trigonometry De Moivre's Theorem Extended Sine

Cosine
Sine
Trigonometry Contents

## Sine

Beginning with DeMoivre's formula: [2.1]

We find an expression for the sine: [1.1]

Expanding the sine and noting that i2=−1, we get: [1.2]
Dividing by i, and grouping the sines: [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): [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 [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: [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: [1.7]

Writing (cos2 x)p-k-1 as 1−sin2 x)p-k-1 and writing the expansion: [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: [1.9]
Where p is the ceiling (n/2), or , 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: [1.10]
Where u=floor(r/2) and p=ceiling(n/2).

Trigonometry Contents

Ken Ward's Mathematics Pages

# Faster Arithmetic - by Ken Ward

Ken's book is packed with examples and explanations that enable you to discover more than 150 techniques to speed up your arithmetic and increase your understanding of numbers. Paperback and Kindle: 