Ken Ward's Mathematics Pages

Trigonometry Sine Multiple Angle Formulae

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

Page Contents

1. Formulae
2. Table of Sines n=0, n<=10
3. Observations from the table
4. Proof
1. Relationship Between Terms n is even
2. Relationship between terms when n is odd
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 

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

4. 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. 
5. The powers are all odd.
6. 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.
7. The final term's coefficient is 2^n-1. 
8. The number of terms is ceiling [(n)/2] , for instance for n=5, the number of terms is ceiling ( (5)/2)=3
9. 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 , where k is term 0, 1... The formula relate to n>1 and k>0. 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:

        [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]
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Relationship between terms when n is odd

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:
[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 cos² x, c², with Equation 5.5 in 5.4:
[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]

■

The first term coefficient is n

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

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.

■



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