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].

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This applet requires Sun's Java VM 2 which your browser may perceive as a popup. Which it is not. If you want to see the applet work, visit Sun's website at http://www.java.com/en/download/index.jsp, download and install Java VM and enjoy the applet.

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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 A, 90° - A, B, 90° - B. The vertices of the rhombus split the sides of the rectangle into segments of lengths cos A, sin A, cos B, sin B, as shown.

The area of the rhombus is sin(A + B). In the right half of the applet, the triangles rearranged leaving two rectangles unoccupied. The area of one is sin A × cos B, that of the other cos A × sin B, proving the "sine of the sum" formula

sin(A + B) = sinA cosB + cosA sinB.

References

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

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  • Addition and Subtraction Formulas for Sine and Cosine III
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• Hunting Right Angles
• Point on Bisector in Right Angle
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Proofs Without Words

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• Ceva's Theorem: Proof Without Words
• Viviani's Theorem
• A Property of Rhombi
• Triangular Numbers in a Square
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