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)
Title addition and subtraction formulas for sine and cosine
Canonical name AdditionAndSubtractionFormulasForSineAndCosine
Date of creation 2013-03-22 16:59:01
Last modified on 2013-03-22 16:59:01
Owner Wkbj79 (1863)
Last modified by Wkbj79 (1863)
Numerical id 16
Author Wkbj79 (1863)
Entry type Derivation
Classification msc 26A09
Classification msc 15-00
Classification msc 33B10
Synonym addition and subtraction formulae for sine and cosine
Synonym addition formulas for sine and cosine
Synonym addition formulae for sine and cosine
Synonym subtraction formulas for sine and cosine
Synonym subtraction formulae for sine and cosine
Synonym addition formula for sine
Synonym subtraction
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