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Multiple-Angle Formulas

For a positive integer, expressions of the form , , and can be expressed in terms of and only using the Euler formula and binomial theorem.

For ,

			(1)
			(2)
			(3)
			(4)
			(5)
			(6)
			(7)

Particular cases for multiple angle formulas for are given by

			(8)
			(9)
			(10)
			(11)
			(12)

The function can also be expressed as a polynomial in (for odd) or times a polynomial in as

$sin(nx)={(-1)^((n-1)/2)T_n(sinx) for n odd; (-1)^(n/2-1)cosxU_(n-1)(sinx) for n even,$

(13)

where is a Chebyshev polynomial of the first kind and is a Chebyshev polynomial of the second kind. The first few cases are

			(14)
			(15)
			(16)
			(17)

Similarly, can be expressed as times a polynomial in as

(18)

The first few cases are

			(19)
			(20)
			(21)
			(22)

Bromwich (1991) gave the formula

$sin(na)={nx-(n(n^2-1^2)x^3)/(3!)+(n(n^2-1^2)(n^2-3^2)x^5)/(5!)-... for n odd; ncosa[x-((n^2-2^2)x^3)/(3!)+((n^2-2^2)(n^2-4^2)x^5)/(5!)-...] for n even,$

(23)

where .

For , the multiple-angle formula can be derived as

			(24)
			(25)
			(26)
			(27)
			(28)
			(29)
			(30)

The first few values are

			(31)
			(32)
			(33)
			(34)
			(35)

The function can also be expressed as a polynomial in (for even) or times a polynomial in as

$cos(nx)={(-1)^((n-1)/2)cosxU_(n-1)(sinx) for n odd; (-1)^(n/2)T_n(sinx) for n even.$

(36)

The first few cases are

			(37)
			(38)
			(39)
			(40)

Similarly, can be expressed as a polynomial in as

(41)

The first few cases are

			(42)
			(43)
			(44)
			(45)

Bromwich (1991) gave the formula

$cos(na)={cosa[1-((n^2-1^2)x^2)/(2!)+((n^2-1^2)(n^2-3^2)x^4)/(4!)-...] n odd; 1-(n^2x^2)/(2!)+(n^2(n^2-2^2)x^4)/(4!)-... n even,$

(46)

where .

The first few multiple-angle formulas for are

			(47)
			(48)
			(49)

are given by Beyer (1987, p. 139) for up to .

Multiple-angle formulas can also be written using the recurrence relations

			(50)
			(51)
			(52)

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