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geometric derivation of addition formulas for sine and cosine
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(Derivation)
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Here is a geometric derivation of the addition laws for sines (and cosines)
First note that, by symmetry, it is clear that
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:
Now, from the definitions of $\sin$ and $\cos$ , and assuming that $OA=1$ , we see that
Now,
so we have that
But also, it is clear that $\angle BCA=\angle DOC=\alpha$ , and therefore we have similarly
so that
Thus $\sin(\alpha+\beta)$ is the $y$ -coordinate of $A$ , which is the $y$ -coordinate of $B$ , so $$ \sin(\alpha+\beta)=CD+BC=\sin(\alpha)\cos(\beta)+\cos(\alpha)\sin(\beta $$ 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 $$ \cos(\alpha+\beta)=OD-AB=\cos(\alpha)\cos(\beta)-\sin(\alpha)\sin(\beta $$
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"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
There is 1 reference to this entry.
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 2129 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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Pending Errata and Addenda
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