The formula for sin.(alpha + beta)

The formula for sin. $\left( {\alpha + \beta }\right)$

We consider a triangle with sides $a, b, c$ and opposite angles $\alpha, \beta, \gamma$ respectively:

We have added the altitude CF; the additional annotation follows from the definitions of the sine and cosine functions. We observe

true 1 the two annotations for CF} a.sin.β = b.sin.x [\text{a : }b = \sin \alpha : \sin \beta ] $\equiv \left{ \begin{array}{l} \text{symmetric } \ a : b : c = \sin \alpha : \sin \beta : \sin \gamma \end{array}\right.$ (*)

Next we observe

true

[\left{ \begin{array}{l} \text{annotations for BF and FA}} \end{array}\right. ]

$$ \begin{array}{l} c = a \cdot \cos . \beta + b \cdot \cos . \alpha \ \equiv {() } \ \sin \gamma = \sin \alpha \cdot \cos \beta + \sin \beta \cdot \cos \alpha \ \equiv \quad {x + \beta + \gamma = \pi } \ \sin . (\alpha + \beta) = \sin . \alpha \cdot \cos . \beta + \sin . \beta \cdot \cos . \alpha (x _ {}) \ \end{array} $$

and so we have proved the addition formula $(\ast \ast)$ for the sine function for $0 \leq \alpha, 0 \leq \beta$ and $\alpha + \beta \leq \pi$ . (Note that $F$ does not need to lie between $A$ and $B$ for $(f)$ to be valid.)

Austin. 11 September 1996

prof. dr. Edsger W.Dgkstra Department of Computer Sciences The University of Texas at Austin Austin, TX 78712-1188 USA