derive multiple angle formula
Answers
Answer:
For n a positive integer, expressions of the form sin(nx), cos(nx), and tan(nx) can be expressed in terms of sinx and cosx only using the Euler formula and binomial theorem.
For sin(nx),
sin(nx) = (e^(inx)-e^(-inx))/(2i)
(1)
= ((e^(ix))^n-(e^(-ix))^n)/(2i)
(2)
= ((cosx+isinx)^n-(cosx-isinx)^n)/(2i)
(3)
= sum_(k=0)^(n)(n; k)(cos^kx(isinx)^(n-k)-cos^kx(-isinx)^(n-k))/(2i)
(4)
= sum_(k=0)^(n)(n; k)cos^kxsin^(n-k)x(i^(n-k)-(-i)^(n-k))/(2i)
(5)
= sum_(k=0)^(n)(n; k)cos^kxsin^(n-k)xsin[1/2(n-k)pi].
(6)
The first few values are given by
sin(2x) = 2cosxsinx
(7)
sin(3x) = 3cos^2xsinx-sin^3x
(8)
sin(4x) = 4cos^3xsinx-4cosxsin^3x
(9)
sin(5x) = 5cos^4xsinx-10cos^2xsin^3x+sin^5x.
(10)
Other related formulas include
sin(nx) = sinxsum_(k=0)^(|_(n-1)/2_|)(-1)^k(n-k-1; k)2^(n-2k-1)cos^(n-2k-1)x
(11)
= sum_(k=0)^(|_(n-1)/2_|)(-1)^k(n; 2k+1)sin^(2k+1)xcos^(n-2k-1)x,
(12)
where |_x_| is the floor function.
A product formula for sin(nx) is given by
sin(nx)=2^(n-1)product_(k=0)^(n-1)sin((pik)/n+x).
(13)
The function sin(nx) can also be expressed as a polynomial in sinx (for n odd) or cosx times a polynomial in sinx 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 T_n is a Chebyshev polynomial of the first kind and U_n is a Chebyshev polynomial of the second kind. The first few cases are
sin(2x) = 2cosxsinx
(15)
sin(3x) = 3sinx-4sin^3x
(16)
sin(4x) = cosx(4sinx-8sin^3x)
(17)
sin(5x) = 5sinx-20sin^3x+16sin^5x.
(18)
Similarly, sin(nx) can be expressed as sinx times a polynomial in cosx as
sin(nx)=sinxU_(n-1)(cosx).
(19)
The first few cases are
sin(2x) = 2cosxsinx
(20)
sin(3x) = sinx(-1+4cos^2x)
(21)
sin(4x) = sinx(-4cosx+8cos^3x)
(22)
sin(5x) = sinx(1-12cos^2x+16cos^4x).
(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 x=sina.
For cos(nx), the multiple-angle formula can be derived as
cos(nx) = (e^(inx)+e^(-inx))/2
(25)
= ((e^(ix))^n+(e^(-ix))^n)/2
(26)
= ((cosx+isinx)^n+(cosx-isinx)^n)/2
(27)
= sum_(k=0)^(n)(n; k)(cos^kx(isinx)^(n-k)+cos^kx(-isinx)^(n-k))/2
(28)
= sum_(k=0)^(n)(n; k)cos^kxsin^(n-k)x(i^(n-k)+(-i)^(n-k))/2
(29)
= sum_(k=0)^(n)(n; k)cos^kxsin^(n-k)xcos[1/2(n-k)pi].
(30)
The first few values are
cos(2x) = cos^2x-sin^2x
(31)
cos(3x) = cos^3x-3cosxsin^2x
(32)
cos(4x) = cos^4x-6cos^2xsin^2x+sin^4x
(33)
cos(5x) = cos^5x-10cos^3xsin^2x+5cosxsin^4x.
(34)
Other related formulas include
cos(nx) = nsum_(k=0)^(|_n/2_|)((-1)^k(n-k-1)!2^(n-2k-1)cos^(n-2k)x)/(k!(n-2k!))
(35)
= 2^(n-1)cos^nx+nsum_(k=1)^(|_n/2_|)((-1)^k)/k(n-k-1; k-1)2^(n-2k-1)cos^(n-2k)x
(36)
= sum_(k=0)^(|_n/2_|)(-1)^k(n; 2k)sin^(2k)xcos^(n-2k)x.
(37)
The function cos(nx) can also be expressed as a polynomial in sinx (for n even) or cosx times a polynomial in sinx 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
cos(2x) = 1-2sin^2x
(39)
cos(3x) = cosx(1-4sin^2x)
(40)
cos(4x) = 1-8sin^2x+8sin^4x
(41)
cos(5x) = cosx(1-12sin^2x+16sin^4x).
(42)
Similarly, cos(nx) can be expressed as a polynomial in cosx as
cos(nx)=T_n(cosx).
(43)
The first few cases are
cos(2x) = -1+2cos^2x
(44)
cos(3x) = -3cosx+4cos^3x
(45)
cos(4x) = 1-8cos^2x+8cos^4x
(46)
cos(5x) = 5cosx-20cos^3x+16cos^5x.
(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 x=sina.
The first few multiple-angle formulas for tan(nx) are
tan(2x) = (2tanx)/(1-tan^2x)
(49)
tan(3x) = (3tanx-tan^3x)/(1-3tan^2x)
(50)
tan(4x) = (4tanx-4tan^3x)/(1-6tan^2x+tan^4x)
(51)
are given by Beyer (1987, p. 139) for up to n=6.
Multiple-angle formulas can also be written using the recurrence relations
sin(nx) = 2sin[(n-1)x]cosx-sin[(n-2)x]
(52)
cos(nx) = 2cos[(n-1)x]cosx-cos[(n-2)x]
(53)
tan(nx) = (tan[(n-1)x]+tanx)/(1-tan[(n-1)x]tanx).
Step-by-step explanation:
hope it helps