Generating Sin(nx) and Cos(nx)
By Zuhair Omari
How to express Sin(nx) and Cos(nx) in terms of only Sinx and integer n, or only Cosx and integer n ? Such as:
For quick look, here are the final formulas:
…………(1)
…………..(2)
….....……(3)
..…….(4)
…………….(5)
……..…….(6)
………….(7)
…………(8)
Which we called " Omari's formulas ".
And the coefficients:
are " Omari's coefficients ".
Some examples for calculating Omari's coefficients:
Example 1:
= =576
Example 2:
The next tables give the values of Omari's coefficients for the integer n up to 5.
Ank
k 0 1 2 3 4 5
n 2n+1
0 1 1
1 3 3 4
2 5 5 20 16
3 7 7 56 112 64
4 9 9 120 432 576 256
5 11 11 220 1232 2816 2816 1024
Bnk
k 0 1 2 3 4 5
n 2n+1
0 1 1
1 3 1 4
2 5 1 12 16
3 7 1 24 80 64
4 9 1 40 240 448 256
5 11 1 60 560 1792 2304 1024
Cnk
k 0 1 2 3 4 5
n 2n
1 2 2
2 4 4 8
3 6 6 32 32
4 8 8 80 192 128
5 10 10 160 672 1024 512
Dnk
k 0 1 2 3 4 5
n 2n
1 2 1 2
2 4 1 8 8
3 6 1 18 48 32
4 8 1 32 160 256 128
5 10 1 50 400 1120 1280 512
How to get Omari's formulas
We will start from Moivere's formula:
…..(9)
To define the terms in (9) if they are real or imaginary, we must define n is it even or odd. Therefore, we will use 2n for even, and 2n+1 for odd. We will get two formulas:
…(10 )
…………(11)
After separating real parts and imaginary, formulas (10) and (11) will give four formulas. We will continue with one of them. The real part of formula (10) is:
…..(12 )
This formula is already known to experts, as will as the other three. Our goal is to have in the right side terms with sinx only or cosx only. Therefore, we need to eliminate sinx once, and cosx in the second time by using the identity: , so we have eight formulas totally. Here, we will eliminate sinx:
.….(13)
Now, we have to rearrange terms so as to put together terms with the same exponent of cosx:
…..(8)
we got formula (8). The other formulas (1),…(7) could be got in the same way.
These formulas, in the first time, were put in empirical way, and were proved with means of "mathematical induction" method; later, they were developed from known mathematical formulas, as we saw. So, the proof with means of "mathematical induction" exists.
Eng. Zuhair Omari
Registered in Jordan, at the Ministry of Culture- Department of National Library, under No. 860/3/2008. In March 25. 2008.