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[parent] geometric derivation of addition formulas for sine and cosine (Derivation)

Here is a geometric derivation of the addition laws for sines (and cosines)

$\displaystyle \sin(\alpha+\beta)$ $\displaystyle =\sin(\alpha)\cos(\beta)+\cos(\alpha)\sin(\beta)$    
$\displaystyle \cos(\alpha+\beta)$ $\displaystyle =\cos(\alpha)\cos(\beta)-\sin(\alpha)\sin(\beta)$    

First note that, by symmetry, it is clear that

$\displaystyle \sin\left(x+\frac{\pi}{2}\right)$ $\displaystyle =\cos x$    
$\displaystyle \cos\left(x+\frac{\pi}{2}\right)$ $\displaystyle =-\sin x$    
$\displaystyle \sin(-x)$ $\displaystyle =-\sin x$    
$\displaystyle \cos(-x)$ $\displaystyle =\cos x$    

and so we can reduce proving the addition law to proving it in the case where $ \alpha$, $ \beta$, and $ \alpha+\beta$ are all in the first quadrant.

We then have the situation pictured below:


\begin{pspicture}(-.4,-.4)(5.4,5.4) \psline[linewidth=1pt]{->}(-.4,0)(5.4,0) \ps... ...3){A} \rput(3.88,4.63){B} \rput(4.08,2.04){C} \rput(3.88,-.3){D} \end{pspicture}

Now, from the definitions of $ \sin$ and $ \cos$, and assuming that $ OA=1$, we see that

$\displaystyle AC$ $\displaystyle =\sin(\beta)$    
$\displaystyle OC$ $\displaystyle =\cos(\beta)$    

Now,

$\displaystyle \sin(\alpha)$ $\displaystyle =\sin(\angle DOC)=\frac{CD}{OC}=\frac{CD}{\cos(\beta)}$    
$\displaystyle \cos(\alpha)$ $\displaystyle =\cos(\angle DOC)=\frac{OD}{OC}=\frac{OD}{\cos(\beta)}$    

so we have that
$\displaystyle CD$ $\displaystyle =\sin(\alpha)\cos(\beta)$    
$\displaystyle OD$ $\displaystyle =\cos(\alpha)\cos(\beta)$    

But also, it is clear that $ \angle BCA=\angle DOC=\alpha$, and therefore we have similarly

$\displaystyle \sin(\alpha)$ $\displaystyle =\sin(\angle BCA)=\frac{AB}{AC}=\frac{AB}{\sin(\beta)}$    
$\displaystyle \cos(\alpha)$ $\displaystyle =\cos(\angle BCA)=\frac{BC}{AC}=\frac{BC}{\sin\beta}$    

so that
$\displaystyle AB$ $\displaystyle =\sin(\alpha)\sin(\beta)$    
$\displaystyle BC$ $\displaystyle =\cos(\alpha)\sin(\beta)$    

Thus $ \sin(\alpha+\beta)$ is the $ y$-coordinate of $ A$, which is the $ y$-coordinate of $ B$, so
$\displaystyle \sin(\alpha+\beta)=CD+BC=\sin(\alpha)\cos(\beta)+\cos(\alpha)\sin(\beta)$
and $ \cos(\alpha+\beta)$ is the $ x$-coordinate of $ A$, which is the $ x$-coordinate of $ B$ less the difference in $ x$-coordinates between $ B$ and $ A$, so
$\displaystyle \cos(\alpha+\beta)=OD-AB=\cos(\alpha)\cos(\beta)-\sin(\alpha)\sin(\beta)$



"geometric derivation of addition formulas for sine and cosine" is owned by rm50.
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Cross-references: difference, definitions, quadrant, clear, symmetry, cosines, sines, addition, derivation
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This is version 4 of geometric derivation of addition formulas for sine and cosine, born on 2007-05-28, modified 2007-05-29.
Object id is 9483, canonical name is GeometricDerivationOfAdditionFormulasForSineAndCosine.
Accessed 1025 times total.

Classification:
AMS MSC: 15-00 (Linear and multilinear algebra; matrix theory :: General reference works )
 26A09 (Real functions :: Functions of one variable :: Elementary functions)
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