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Ken Ward's Mathematics Pages

Trigonometry Cosine Multiple Angle Formulae

This page lists the formulae for cos nx for n=2, up to n=10. By studying the table of coefficients of these formulae, we can infer other formulae. For instance, relationshipBetweenTermsForCosine.gif, relating the terms in the table, and enabling us to compute further formulae in an easier manner.
See also:

Trigonometry Contents

Page Contents

  1. Formulae
  2. Table of Cosines n=0, n<=10
  3. Observations from the table
  4. Proof
    1. First term coefficient is 2n-1cosineT0.gif
    2. Relationship between the terms relationshipBetweenTermsForCosine.gif


Formulae

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

Table of Cosines n=0, n<=10

The following table shows the coefficients of the cosine 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.
k012345
Power n n-2 n-4 n-6 n-8 n-10Formulae
n0 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

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, and they aren't that difficult to understand! We note the following:
  1. Beginning with n=2, the first term coefficients are double the previous one. So, writing T to represent a term, and Tnk.gif  to refer to the term at line n and column k of the table, with k being 0, 1, ... we note that:
    cosineT0.gif, that is, the first term coefficient doubles each time. Also, for n>0, the first term is 2n-1
  2. We can also note that 
        relationshipBetweenTermsForCosine.gif,
        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. 
  1. 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. 
  2. The powers are either all even or all odd.
  3. 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). 
  4. The number of terms is ceiling [(n+1)/2] (ceil(nAnd1)Over2.gif), for instance for n=5, the number of terms is ceiling ( (5+1)/2)=3

Proof

Let us refer to the polynomial representing cos nx as Tn, each term, k as chebyshevTn_m.gif, where k is term 0, 1... The formula relate to n>1.

We can write cos nx, as Tn, also abbreviating cos x as c, as we would anyway, in our notebooks:
chebyshevTerms1.gif[4.1]
And the previous two terms, representing cos (n-1)x and cos (n-2)x as:
chebyshevTerms2.gif [4.2]

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

chebyshevTerms3.gif [4.3]

Using 4.3 applied to equations 4.2, we have:
chebyshevTerms4.gif [4.3]
Note that the coefficient for term k-1 n-2, chebyshevTermk_1.gif,has been added to match up the powers.

Adding like terms, we have:
chebyshevTerms5.gif [4.4]

First term coefficient is 2n-1

From Equation 4.4, above, we can deduce that the coefficient for the first term, k=0, is double the previous one. Because cos (1x)=cos x (the first coefficient is 1, when n=1), by the formula 4.4, each subsequent first term coefficient is double, the previous one. So the first term coefficient for cos 2x is 2, and the first term coefficient for cos(3x) is 4, and so the nth one is 2n-1.


Relationship between the terms relationshipBetweenTermsForCosine.gif

From Equation 4.4, the general term coefficient is precisely our formula:
relationshipBetweenTermsForCosine.gif

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




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