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Trigonometry Sine Multiple Angle Formulae

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Trigonometry Contents

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  1. Formulae
  2. Table of Sines n=0, n<=10
  3. Observations from the table
  4. Proof
    1. Relationship Between Terms n is evenchebyshevsine_even_n_k.gif
    2. Relationship between terms when n is oddchebyshevsine_odd_n_k.gif
    3. The first term coefficient is n


List of Formulae

[1.1]
[1.2]
[1.3]
[1.4]
[1.5]
[1.6]
[1.7]
[1.8]
[1.9]

Table of Sines n=0, n<=10

The following table shows the coefficients of the sine polynomials for the appropriate multiple angle. The top of the table shows the powers, which begin with n and decrease by two for each additional term, if any. The formulae on the right are meant to clarify the table. The number of each term is k=0, 1, 2...
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 .

Observations from the table

What follows are observations, not all of which are proved on this page. At this stage, they are hypotheses (if we are thinking scientically about the data in the table) or conjectures (if we are thinking mathematically). [They are all provable, however]
The patterns in the table are quite interesting. We note the following:
  1. The first term coefficients are equal to n. 
  2. There appear to be two patterns, one for even n and one for odd n.
  3. We can also note that 
        chebyshevsine_odd_n_k.gif
        and 
        chebyshevsine_even_n_k.gif
        Actually, these formula were inferred, rather than observed, by picking out the powers of sine from the Chebyshev Method.
  1. 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. 
  2. The powers are all odd.
  3. 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.
  4. The final term's coefficient is 2n-1
  5. The number of terms is ceiling [(n)/2] , for instance for n=5, the number of terms is ceiling ( (5)/2)=3
  6. When n is odd, the coefficients for sin nx are ± the same numbers appearing for the cosine. 
        For instance:
       
       

Proof

Relationship Between Terms n is even

Let us refer to the polynomial representing sin nx as Tn, and the coefficient of each term, k as chebyshevTn_m.gif, where k is term 0, 1... The formula relate to n>1 and k>0. chebyshevTn_m.gif refers to the numbers in the table above, with n referring to the rows and k to the columns. In point 3 above, we inferred:

        chebyshevsine_even_n_k.gif [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
coefficientsSin1.gif [4.2]
coefficientsSin2.gif [4.3]
coefficientsSin3.gif [4.4]

According to Chebyshev [cos nx=2·cos(n-1)x-cos(n-2)x]:
coefficientsSin4.gif [4.5]

So, using Equations 4.3 and 4.4:
coefficientsSin5.gif [4.6]

Combining powers of sine:
coefficientsSin6.gif [4.7]

Equating powers of sine for the general terms in Equation 4.2 and 4.7, we find our formula:
coefficientsSin7.gif [4.8]


Relationship between terms when n is oddchebyshevsine_odd_n_k.gif

Using the same letters as before, to represent our functions and coefficients, we note that when n is odd, the terms are pure sines. Therefore, we can write our functions as:
stOdd.gif [5.1]
stOdd2.gif [5.2]
stOdd3.gif [5.3]
According to Chebyshev [cos nx=2·cos(n-1)x-cos(n-2)x]:
coefficientsSin4.gif

Multiplying Equation 5.2 by 2c, we get:
stOdd4.gif [5.4]
Because:
stOdd5.gif [5.5]

We can substitute for cos2 x, c2, with Equation 5.5 in 5.4:
stOdd6.gif [5.5]

Combining the coefficients of like powers:
stOdd7.gif [5.6]
To complete our work, we now need to subtract Equation 5.3 from 5.6, to complete our Chebyshev relationship.

stOdd8.gif [5.7]

Collecting like terms in Equation 5.7
stOdd9.gif [5.8]

By equating the general terms (same powers of sine) in Equation 5.1 and Equation 5.8, we obtain our equation:
stOdd10.gif [5.9]


The first term coefficient is n

Whether n is odd or even, the first term (in Equations 4.7 and 5.8) is: stOdd11.gif [6.1]

That is, in terms of the coefficients:
stFirst.gif [6.2]

Rearranging,
stFirst3.gif [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:
stFirst2.gif [6.4]

That is,
stFirst4.gif [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, tn0.gif=1=n. When we increment n by 1, we also increment tn0.gif by 1 (by Equation 6.5), so tn0.gif remains equal to n for all n.





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