See also:

- De Moivre's Theorem
- Chebyshev's Method
- Multiple Angles Cosines
- Multiple Angles Sines
- Multiple Angles Tangents
- De Moivre's Theorem Extended

Trigonometry Contents

[1.2]

[1.3]

[1.4]

[1.5]

[1.6]

[1.7]

[1.8]

[1.9]

k | 0 | 1 | 2 | 3 | 4 | 5 | ||

Power | n | n-2 | n-4 | n-6 | n-8 | n-10 | Formulae | |
---|---|---|---|---|---|---|---|---|

n | 0 | 1 | . | . | . | . | . | cos (0x)=1 |

1 | 1 | . | . | . | . | . | cos(1x)=cos x | |

2 | 2 | -1 | . | . | . | . | ||

3 | 4 | -3 | . | . | . | . | ||

4 | 8 | -8 | 1 | . | . | . | ||

5 | 16 | -20 | 5 | . | . | . | ||

6 | 32 | -48 | 18 | -1 | . | . | ||

7 | 64 | -112 | 56 | -7 | . | . | ||

8 | 128 | -256 | 160 | -32 | 1 | . | ||

9 | 256 | -576 | 432 | -120 | 9 | . | ||

10 | 512 | -1280 | 1120 | -400 | 50 | -1 |

The patterns in the table are quite interesting, and they aren't that difficult to understand! We note the following:

- Beginning with n=2, the first term coefficients are double the previous one. So, writing T to represent a term, and to refer to the term at line n and column k of the table, with k being 0, 1, ... we note that:

, that is, the first term coefficient doubles each time. Also, for n>0, the first term is 2^{n-1}. - We can also note that

for k>0 and n>1. For instance, when n=5, and k=1, we find the term is 2·(-20)-8, which is -48, where -20 is term 1 when n=4, and 8 is term 0 when n=3.

- With the cosine (opposite that for the sine), the highest power is always positive, which is why we write the formula and the table from the highest power down, to avoid beginning with a minus sign (there is, of course no mathematical reason to do this!). The subsequent terms alternate in sign; for instance, term 1 is always negative, term 2, always positive, etc.
- The powers are either all even or all odd.
- The final term's coefficient is either ±1 (when the power of the cosine is 0) or ±n (when the power of the cosine is 1).
- The number of terms is ceiling [(n+1)/2] (), for instance for n=5, the number of terms is ceiling ( (5+1)/2)=3

We can write cos nx, as T

[4.1]

And the previous two terms, representing cos (n-1)x and cos (n-2)x as:

[4.2]

Using the Chebyshev Method, we can relate 4.1 and 4.2:

[4.3]

Using 4.3 applied to equations 4.2, we have:

[4.3]

Note that the coefficient for term k-1 n-2, ,has been added to match up the powers.

Adding like terms, we have:

[4.4]

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The other observations aren't proved on this page. They can be proved using DeMoivre's theorem and the binomial theorem.

Trigonometry Contents

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