Multiple-Angle Formulas -- from Wolfram MathWorld

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)

The first few values are given by

			(7)
			(8)
			(9)
			(10)

Other related formulas include

			(11)
			(12)

where is the floor function.

A product formula for is given by

(13)

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

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

(14)

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

			(15)
			(16)
			(17)
			(18)

Similarly, can be expressed as times a polynomial in as

(19)

The first few cases are

			(20)
			(21)
			(22)
			(23)

Bromwich (1991) gave the formula

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

(24)

where .

For , the multiple-angle formula can be derived as

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

The first few values are

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

Other related formulas include

			(35)
			(36)
			(37)

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

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

(38)

The first few cases are

			(39)
			(40)
			(41)
			(42)

Similarly, can be expressed as a polynomial in as

(43)

The first few cases are

			(44)
			(45)
			(46)
			(47)

Bromwich (1991) gave the formula

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

(48)

where .

The first few multiple-angle formulas for are

			(49)
			(50)
			(51)

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

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

			(52)
			(53)
			(54)

Multiple-Angle Formulas

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References

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