# 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].

What if applet does not run? |

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,

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 + B) = sinA cosB + cosA sinB.

### References

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

### Trigonometry

- What Is Trigonometry?
- Addition and Subtraction Formulas for Sine and Cosine
- The Law of Cosines (Cosine Rule)
- Cosine of 36 degrees
- Tangent of 22.5
^{o}- Proof Wthout Words - Sine and Cosine of 15 Degrees Angle
- Sine, Cosine, and Ptolemy's Theorem
- arctan(1) + arctan(2) + arctan(3) = π
- Trigonometry by Watching
- arctan(1/2) + arctan(1/3) = arctan(1)
- Morley's Miracle
- Napoleon's Theorem
- A Trigonometric Solution to a Difficult Sangaku Problem
- Trigonometric Form of Complex Numbers
- Derivatives of Sine and Cosine
- ΔABC is right iff sin²A + sin²B + sin²C = 2
- Advanced Identities
- Hunting Right Angles
- Point on Bisector in Right Angle
- Trigonometric Identities with Arctangents
- The Concurrency of the Altitudes in a Triangle - Trigonometric Proof
- Butterfly Trigonometry
- Binet's Formula with Cosines
- Another Face and Proof of a Trigonometric Identity
- cos/sin inequality
- On the Intersection of kx and |sin(x)|
- Cevians And Semicircles
- Double and Half Angle Formulas
- A Nice Trig Formula
- Another Golden Ratio in Semicircle
- Leo Giugiuc's Trigonometric Lemma
- Another Property of Points on Incircle
- Much from Little
- The Law of Cosines and the Law of Sines Are Equivalent
- Wonderful Trigonometry In Equilateral Triangle
- A Trigonometric Observation in Right Triangle

### Proofs Without Words

- Proofs Without Words
- Sums of Geometric Series - Proofs Without Words
- Sine of the Sum Formula
- Parallelogram Law: A PWW
- Parallelogram Law
- Ceva's Theorem: Proof Without Words
- Viviani's Theorem
- A Property of Rhombi
- Triangular Numbers in a Square
- PWW: How Geometry Helps Algebra
- Varignon's Theorem, Proof Without Words

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