Geometry > Trigonometry > Trigonometric Identities >

Multiple-Angle Formulas

DOWNLOAD Mathematica Notebook

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).
(54)

Wolfram Web Resources

Mathematica »

The #1 tool for creating Demonstrations and anything technical.

 Wolfram|Alpha »

Explore anything with the first computational knowledge engine.

 Wolfram Demonstrations Project »

Explore thousands of free applications across science, mathematics, engineering, technology, business, art, finance, social sciences, and more.
Computerbasedmath.org »

Join the initiative for modernizing math education.

 Online Integral Calculator »

Solve integrals with Wolfram|Alpha.

 Step-by-step Solutions »

Walk through homework problems step-by-step from beginning to end. Hints help you try the next step on your own.
Wolfram Problem Generator »

Unlimited random practice problems and answers with built-in Step-by-step solutions. Practice online or make a printable study sheet.

 Wolfram Education Portal »

Collection of teaching and learning tools built by Wolfram education experts: dynamic textbook, lesson plans, widgets, interactive Demonstrations, and more.

 Wolfram Language »

Knowledge-based programming for everyone.