addition and subtraction formulas for sine and cosine

The rotation matrix $\displaystyle\left(\begin{array}[]{lr}\cos\theta&-\sin\theta\\ \sin\theta&\cos\theta\end{array}\right)$ will be used to obtain the addition formulas for sine and cosine.

Recall that a vector in $\mathbb{R}^{2}$ can be rotated $\theta$ radians in the counterclockwise direction by multiplying on the left by the rotation matrix $\displaystyle\left(\begin{array}[]{lr}\cos\theta&-\sin\theta\\ \sin\theta&\cos\theta\end{array}\right)$ . Because rotating by $\alpha+\beta$ radians is the same as rotating by $\beta$ radians followed by rotating by $\alpha$ radians, we obtain:

$\begin{array}[]{rl}\displaystyle\left(\begin{array}[]{lr}\cos(\alpha+\beta)&-% \sin(\alpha+\beta)\\ \sin(\alpha+\beta)&\cos(\alpha+\beta)\end{array}\right)&=\displaystyle\left(% \begin{array}[]{lr}\cos\alpha&-\sin\alpha\\ \sin\alpha&\cos\alpha\end{array}\right)\left(\begin{array}[]{lr}\cos\beta&-% \sin\beta\\ \sin\beta&\cos\beta\end{array}\right)\\ &\\ &=\displaystyle\left(\begin{array}[]{lr}\cos\alpha\cos\beta-\sin\alpha\sin% \beta&-\cos\alpha\sin\beta-\sin\alpha\cos\beta\\ \sin\alpha\cos\beta+\cos\alpha\sin\beta&-\sin\alpha\sin\beta+\cos\alpha\cos% \beta\end{array}\right)\end{array}$

Hence, $\sin(\alpha+\beta)=\sin\alpha\cos\beta+\cos\alpha\sin\beta$ and $\cos(\alpha+\beta)=\cos\alpha\cos\beta-\sin\alpha\sin\beta$ .

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

\sin(\alpha-\beta)=\sin(\alpha+(-\beta))=\sin(\alpha)\cos(-\beta)+\cos(\alpha)% \sin(-\beta)=\sin(\alpha)\cos(\beta)-\cos(\alpha)\sin(\beta)

\cos(\alpha-\beta)=\cos(\alpha+(-\beta))=\cos(\alpha)\cos(-\beta)-\sin(\alpha)% \sin(-\beta)=\cos(\alpha)\cos(\beta)+\sin(\alpha)\sin(\beta)

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
Related topic	AdditionFormula
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Related topic	AdditionFormulas