The formula for sin(alpha + beta)

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

About circles we use the following

Lemma Let arc AB of a circle with diameter d subtend at the centre an angle of $2\varphi$ . Then

(i) at any point P on the remainder of the periphery it subtends an angle (ii) the length of chord AB equals $d \cdot \sin \varphi$

[By choosing $P = {P}^{\prime }$ such that $A{P}^{\prime }$ is a diameter,we make $\angle {AB}{P}^{\prime }$ a right angle and thus see ${AB} = d \cdot \sin \varphi$ .] for the rest of this note we choose $d = 1$ .

NB. ${2\alpha } + {2\beta } = 2\left( {\alpha + \beta }\right)$ .

with the angles at $A$ and $B$ equal to $\alpha$ and $\beta$ respectively,we have according to our lemma

$$ B C = \sin . \alpha A C = \sin . \beta A B = \sin . (\alpha + \beta) $$

and now observe, with CD the altitude on ${AB}$

$$ \begin{array}{l} \sin . (a + b) \ = A B \ = A D + D B \ = A C \cdot \cos . \alpha + B C \cdot \cos . \beta \ = \sin . \beta \cdot \cos . \alpha + \sin . \alpha \cdot \cos . \beta \ \end{array} $$

which establishes the addition formula for $\sin \left( {\alpha + \beta }\right)$ for $0 \leq \alpha ,\beta \leq \pi /2$ .

Austin 10 Sep. 1996

prof. dr. Edsger W. Dgksra Department of Computer Sciences The University of Texas at Austin

Austin, TX 78712 - 1188, USA