- 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 | 1 | 3 | 5 | 7 | 9 | 11 | Formulae | |
---|---|---|---|---|---|---|---|---|

n | 0 | 0 | . | . | . | . | . | sin (0x)=0 |

1 | 1 | . | . | . | . | . | sin(1x)=sin x | |

2 | 2 | . | . | . | . | . | ||

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

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

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

6 | 6 | -32 | 32 | . | . | |||

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

8 | 8 | -80 | 192 | -128 | . | |||

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

10 | 10 | -160 | 672 | -1024 | 512 | . |

The patterns in the table are quite interesting. We note the following:

- The first term coefficients are equal to n.
- There appear to be two patterns, one for even n and one for odd n.
- We can also note that

and

Actually, these formula were inferred, rather than observed, by picking out the powers of sine from the Chebyshev Method.

- With the sine (opposite that for the cosine), the lowest power is always positive. The lowest power of sine in the table is 1. The first term is either sine or sine·cosine. The subsequent terms alternate in sign; for instance, term 1 is always negative, term 2, always positive, etc.
- The powers are all odd.
- When n is odd, the terms consist of sines only; when n is even, each term contains cos x multiplied by a power of sin x.
- The
final term's coefficient is 2
^{n-1}. - The number of terms is ceiling [(n)/2] , for instance for n=5, the number of terms is ceiling ( (5)/2)=3
- When n is odd, the coefficients for sin nx are ± the same numbers appearing for the cosine.

[4.1]

When n is even, all the terms contain one fact of cos x. So the n row has a factor of cosx, the n-1 row is pure sines, and the n-2 row also contains a cos x. We abbreviate cos x as c and sin x as s.

We can write

[4.2]

[4.3]

[4.4]

According to Chebyshev [cos nx=2·cos(n-1)x-cos(n-2)x]:

[4.5]

So, using Equations 4.3 and 4.4:

[4.6]

Combining powers of sine:

[4.7]

Equating powers of sine for the general terms in Equation 4.2 and 4.7, we find our formula:

[4.8]

■

[5.1]

[5.2]

[5.3]

According to Chebyshev [cos nx=2·cos(n-1)x-cos(n-2)x]:

Multiplying Equation 5.2 by 2c, we get:

[5.4]

Because:

[5.5]

We can substitute for cos

[5.5]

Combining the coefficients of like powers:

[5.6]

To complete our work, we now need to subtract Equation 5.3 from 5.6, to complete our Chebyshev relationship.

[5.7]

Collecting like terms in Equation 5.7

[5.8]

By equating the general terms (same powers of sine) in Equation 5.1 and Equation 5.8, we obtain our equation:

[5.9]

■

That is, in terms of the coefficients:

[6.2]

Rearranging,

[6.3]

Noting that the difference between any two terms, (n>1), is a constant, and writing this as "a", and taking another step of noting from the table the difference between any two terms, we find:

[6.4]

That is,

[6.5]

When n=1, then the first term coefficient is 1. As we increment n by 1, we also increment the first term coefficient by 1 (according to Equation 6.5). Therefore the first term is equal to n.

In other words, when n=1, =1=n. When we increment n by 1, we also increment by 1 (by Equation 6.5), so remains equal to n for all n.

■

Trigonometry Contents

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