Sine of the Sum Formula

The applet below illustrates a proof without words of the "sine of the sum" formula due to Volker Priebe and Edgar A. Ramos [Nelsen, p. 40].

The rhombus inscribed into a rectangle has side length of $1.$ The rhombus cuts off of the rectangle two pairs of equal right triangles. The acute angles of the triangles are $\alpha,$ $90^{\circ} -\alpha,$ $\beta,$ $90^{\circ} - \beta.$ The vertices of the rhombus split the sides of the rectangle into segments of lengths $\cos\alpha,$ $\sin\alpha,$ $\cos\beta,$ $\sin\beta,$ as shown.

sine of the sum in rhombus

The area of the rhombus is $\sin(\alpha + \beta).$ In the right half of the applet, the triangles rearranged leaving two rectangles unoccupied. The area of one is $\sin\alpha \times \cos\beta,$ that of the other $\cos\alpha \times \sin\beta,$ proving the "sine of the sum" formula

$\sin(\alpha + \beta) = \sin\alpha \cos\beta + \cos\alpha \sin\beta.$

References

R. B. Nelsen, Proofs Without Words II, MAA, 2000

Trigonometry

Proofs Without Words

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