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单词 AdditionAndSubtractionFormulasForSineAndCosine
释义

addition and subtraction formulas for sine and cosine


The rotation matrixMathworldPlanetmath (cosθ-sinθsinθcosθ) will be used to obtain the addition formulas for sine and cosine.

Recall that a vector in 2 can be rotated θ radians in the counterclockwise direction by multiplying on the left by the rotation matrix (cosθ-sinθsinθcosθ). Because rotating by α+β radians is the same as rotating by β radians followed by rotating by α radians, we obtain:

(cos(α+β)-sin(α+β)sin(α+β)cos(α+β))=(cosα-sinαsinαcosα)(cosβ-sinβsinβcosβ)=(cosαcosβ-sinαsinβ-cosαsinβ-sinαcosβsinαcosβ+cosαsinβ-sinαsinβ+cosαcosβ)

Hence, sin(α+β)=sinαcosβ+cosαsinβ and cos(α+β)=cosαcosβ-sinαsinβ.

Note that sine is an even functionMathworldPlanetmath and that cosine is an odd function, i.e. (http://planetmath.org/Ie) sin(-x)=-sinx and cos(-x)=-cosx. These facts enable us to obtain the subtraction formulas for sine and cosine.

sin(α-β)=sin(α+(-β))=sin(α)cos(-β)+cos(α)sin(-β)=sin(α)cos(β)-cos(α)sin(β)
cos(α-β)=cos(α+(-β))=cos(α)cos(-β)-sin(α)sin(-β)=cos(α)cos(β)+sin(α)sin(β)
Titleaddition and subtraction formulas for sine and cosine
Canonical nameAdditionAndSubtractionFormulasForSineAndCosine
Date of creation2013-03-22 16:59:01
Last modified on2013-03-22 16:59:01
OwnerWkbj79 (1863)
Last modified byWkbj79 (1863)
Numerical id16
AuthorWkbj79 (1863)
Entry typeDerivation
Classificationmsc 26A09
Classificationmsc 15-00
Classificationmsc 33B10
Synonymaddition and subtraction formulae for sine and cosine
Synonymaddition formulas for sine and cosine
Synonymaddition formulae for sine and cosine
Synonymsubtraction formulas for sine and cosine
Synonymsubtraction formulae for sine and cosine
Synonymaddition formula for sine
Synonymsubtraction
Related topicAdditionFormula
Related topicDefinitionsInTrigonometry
Related topicDoubleAngleIdentity
Related topicMeanCurvatureAtSurfacePoint
Related topicDAlembertAndDBernoulliSolutionsOfWaveEquation
Related topicAdditionFormulas
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更新时间:2025/5/4 22:17:52