PlanetMath: angle sum identity

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angle sum identity

(Theorem)

It is desired to prove the identities

$\displaystyle \sin(\theta+\phi) = \sin\theta\cos\phi + \cos\theta\sin\phi$

and

$\displaystyle \cos(\theta+\phi) = \cos\theta\cos\phi - \sin\theta\sin\phi$

Consider the figure

$\includegraphics[width = 5.3cm, height = 5.0cm]{sclaw1.eps}$

where we have

$\circ$: $\triangle Aad \equiv \triangle Ccb$
$\circ$: $\triangle Bba \equiv \triangle Ddc$
$\circ$: .

Also, everything is Euclidean, and in particular, the interior angles of any triangle sum to $\pi$ .

Call $\angle Aad = \theta$ and $\angle baB = \phi$ . From the triangle sum rule, we have $\angle Ada = \frac{\pi}{2}-\theta$ and $\angle Ddc = \frac{\pi}{2} - \phi$ , while the degenerate angle $\angle AdD = \pi$ , so that

$\displaystyle \angle adc = \theta + \phi$

We have, therefore, that the area of the pink parallelogram is $\sin(\theta + \phi)$ . On the other hand, we can rearrange things thus:

$\includegraphics[width = 5.6cm, height=5.1cm]{sclaw2.eps}$

In this figure we see an equal pink area, but it is composed of two pieces, of areas $\sin \phi \cos\theta$ and $\cos\phi \sin\theta$ . Adding, we have

$\displaystyle \sin(\theta + \phi) = \sin\phi\cos\theta + \cos\phi\sin\theta$

which gives us the first. From definitions, it then also follows that $\sin(\theta + \pi/2) = \cos(\theta)$ , and $\sin(\theta+\pi) = - \sin(\theta)$ . Writing

$\displaystyle \begin{array}{rcl} \cos(\theta + \phi)&=& \sin(\theta+\phi+\pi/2)... ...\theta) \cos(\phi) \ &=& \cos\theta\cos\phi - \sin\theta\sin\phi \end{array}$

"angle sum identity" is owned by mathcam. [ full author list (2) | owner history (1) ]

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Cross-references: definitions, parallelogram, area, angle, sum, triangle, interior angles, Euclidean, identities
There are 5 references to this entry.

This is version 11 of angle sum identity, born on 2002-07-17, modified 2005-03-25.
Object id is 3170, canonical name is AngleSumIdentity.
Accessed 5499 times total.

Classification:

AMS MSC:

51-00 (Geometry :: General reference works )

Pending Errata and Addenda

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