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Trigonometric Addition Formulas


Angle addition formulas express trigonometric functions of sums of angles alpha+/-beta in terms of functions of alpha and beta. The fundamental formulas of angle addition in trigonometry are given by

sin(alpha+beta)=sinalphacosbeta+sinbetacosalpha
(1)
sin(alpha-beta)=sinalphacosbeta-sinbetacosalpha
(2)
cos(alpha+beta)=cosalphacosbeta-sinalphasinbeta
(3)
cos(alpha-beta)=cosalphacosbeta+sinalphasinbeta
(4)
tan(alpha+beta)=(tanalpha+tanbeta)/(1-tanalphatanbeta)
(5)
tan(alpha-beta)=(tanalpha-tanbeta)/(1+tanalphatanbeta).
(6)

The first four of these are known as the prosthaphaeresis formulas, or sometimes as Simpson's formulas.

The sine and cosine angle addition identities can be compactly summarized by the matrix equation

 [cosalpha sinalpha; -sinalpha cosalpha][cosbeta sinbeta; -sinbeta cosbeta]=[cos(alpha+beta) sin(alpha+beta); -sin(alpha+beta) cos(alpha+beta)].
(7)

These formulas can be simply derived using complex exponentials and the Euler formula as follows.

cos(alpha+beta)+isin(alpha+beta)=e^(i(alpha+beta))
(8)
=e^(ialpha)e^(ibeta)
(9)
=(cosalpha+isinalpha)(cosbeta+isinbeta)
(10)
=(cosalphacosbeta-sinalphasinbeta)+i(sinalphacosbeta+cosalphasinbeta).
(11)

Equating real and imaginary parts then gives (1) and (3), and (2) and (4) follow immediately by substituting -beta for beta.

Taking the ratio of (1) and (3) gives the tangent angle addition formula

tan(alpha+beta)=(sin(alpha+beta))/(cos(alpha+beta))
(12)
=(sinalphacosbeta+sinbetacosalpha)/(cosalphacosbeta-sinalphasinbeta)
(13)
=((sinalpha)/(cosalpha)+(sinbeta)/(cosbeta))/(1-(sinalphasinbeta)/(cosalphacosbeta))
(14)
=(tanalpha+tanbeta)/(1-tanalphatanbeta).
(15)

The double-angle formulas are

sin(2alpha)=2sinalphacosalpha
(16)
cos(2alpha)=cos^2alpha-sin^2alpha
(17)
=2cos^2alpha-1
(18)
=1-2sin^2alpha
(19)
tan(2alpha)=(2tanalpha)/(1-tan^2alpha).
(20)

Multiple-angle formulas are given by

sin(nx)=sum_(k=0)^(n)(n; k)cos^kxsin^(n-k)xsin[1/2(n-k)pi]
(21)
cos(nx)=sum_(k=0)^(n)(n; k)cos^kxsin^(n-k)xcos[1/2(n-k)pi],
(22)

and can also be written using the recurrence relations

sin(nx)=2sin[(n-1)x]cosx-sin[(n-2)x]
(23)
cos(nx)=2cos[(n-1)x]cosx-cos[(n-2)x]
(24)
tan(nx)=(tan[(n-1)x]+tanx)/(1-tan[(n-1)x]tanx).
(25)
TrigAnglesWeisstein

The angle addition formulas can also be derived purely algebraically without the use of complex numbers. Consider the small right triangle in the figure above, which gives

a=(sinalpha)/(cos(alpha+beta))
(26)
b=sinalphatan(alpha+beta).
(27)

Now, the usual trigonometric definitions applied to the large right triangle give

sin(alpha+beta)=(sinbeta+a)/(cosalpha+b)
(28)
=(sinbeta+(sinalpha)/(cos(alpha+beta)))/(cosalpha+sinalpha(sin(alpha+beta))/(cos(alpha+beta)))
(29)
cos(alpha+beta)=(cosbeta)/(cosalpha+b)
(30)
=(cosbeta)/(cosalpha+sinalpha(sin(alpha+beta))/(cos(alpha+beta))).
(31)

Solving these two equations simultaneously for the variables sin(alpha+beta) and cos(alpha+beta) then immediately gives

sin(alpha+beta)=(cosalphasinalpha+cosbetasinbeta)/(cosalphacosbeta+sinalphasinbeta)
(32)
cos(alpha+beta)=(cos^2beta-sin^2alpha)/(cosalphacosbeta+sinalphasinbeta).
(33)

These can be put into the familiar forms with the aid of the trigonometric identities

 (cosalphacosbeta+sinalphasinbeta)(sinalphacosbeta+sinbetacosalpha)=cosbetasinbeta+cosalphasinalpha
(34)

and

(cosalphacosbeta+sinalphasinbeta)(cosalphacosbeta-sinalphasinbeta)=cos^2alphacos^2beta-sin^2alphasin^2beta
(35)
=1-sin^2alpha-sin^2beta
(36)
=cos^2alpha-sin^2beta
(37)
=cos^2beta-sin^2alpha,
(38)

which can be verified by direct multiplication. Plugging (◇) into (◇) and (38) into (◇) then gives

sin(alpha+beta)=sinalphacosbeta+sinbetacosalpha
(39)
cos(alpha+beta)=cosalphacosbeta-sinalphasinbeta,
(40)

as before.

TrigAdditionSmiley

A similar proof due to Smiley and Smiley uses the left figure above to obtain

 sinalpha=(sin(alpha+beta))/(cosbeta+(sinbetacosalpha)/(sinalpha)),
(41)

from which it follows that

 sin(alpha+beta)=sinalphacosbeta+sinbetacosalpha.
(42)

Similarly, from the right figure,

 (sinalpha)/(cosalpha)=(cosbeta)/(sinbeta+(cos(alpha+beta))/(sinalpha)),
(43)

so

 cos(alpha+beta)=cosalphacosbeta-sinalphasinbeta.
(44)
TrigSubtractionSmiley

Similar diagrams can be used to prove the angle subtraction formulas (Smiley 1999, Smiley and Smiley). In the figure at left,

h=(cosalpha)/(cosbeta)
(45)
x=hsin(alpha-beta)
(46)
=(sinalpha-hsinbeta)cosalpha,
(47)

giving

 sin(alpha-beta)=sinalphacosbeta-cosalphasinbeta.
(48)

Similarly, in the figure at right,

h=(cosalpha)/(sinbeta)
(49)
x=hcos(alpha-beta)
(50)
=(sinalpha+hcosbeta)cosalpha,
(51)

giving

 cos(alpha-beta)=cosalphacosbeta+sinalphasinbeta.
(52)
TanSubtractionRen

A more complex diagram can be used to obtain a proof from the tan(alpha-beta) identity (Ren 1999). In the above figure, let BF/BE=AD/DE. Then

 tan(alpha-beta)=(DE)/(BE)=(AD)/(BF)=(tanalpha-tanbeta)/(1+tanalphatanbeta).
(53)

An interesting identity relating the sum and difference tangent formulas is given by

(tan(alpha-beta))/(tan(alpha+beta))=(sin(alpha-beta)cos(alpha+beta))/(cos(alpha-beta)sin(alpha+beta))
(54)
=((sinalphacosbeta-sinbetacosalpha)(cosalphacosbeta-sinalphasinbeta))/((cosalphacosbeta+sinalphasinbeta)(sinalphacosbeta+sinbetacosalpha))
(55)
=(sinalphacosalpha-sinbetacosbeta)/(sinalphacosalpha+sinbetacosbeta).
(56)

See also

Double-Angle Formulas, Half-Angle Formulas, Harmonic Addition Theorem, Multiple-Angle Formulas, Prosthaphaeresis Formulas, Trigonometry Angles, Trigonometry, Werner Formulas Explore this topic in the MathWorld classroom

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References

Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, 1987.Nelson, R. To appear in College Math. J., March 2000.Ren, G. "Proof without Words: tan(alpha-beta)." College Math. J. 30, 212, 1999.Smiley, L. M. "Proof without Words: Geometry of Subtraction Formulas." Math. Mag. 72, 366, 1999.Smiley, L. and Smiley, D. "Geometry of Addition and Subtraction Formulas." http://math.uaa.alaska.edu/~smiley/trigproofs.html.

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Trigonometric Addition Formulas

Cite this as:

Weisstein, Eric W. "Trigonometric Addition Formulas." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/TrigonometricAdditionFormulas.html

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