Sines and Cosines, Part III (Addition Formulas)

Video Segments

Sines and chord lengths
Addition formula for sines
Ptolemy's theorem on quadrilaterals
Applications of the addition formulas
Sines and cosines of special angles
Application to simple harmonic motion
Preview of part IV
Alexandria--Center of Hellenistic Culture


Sines and Cosines, Part III (Addition Formulas) relates the sine and cosine of an angle with lengths of chords of a circle, as expounded in Claudius Ptolemy's Almagest. This leads to simple derivations of the addition formulas for determining the sine and cosine of a sum of two angles. One application shows that a combination of a sine wave with a cosine wave of the same frequency is another sine wave, possibly shifted. This property plays an important role in the study of simple harmonic motion.

The program also outlines a brief history of the city of Alexandria, founded by Alexander the Great in 331 B.C. His successors created a center of Hellenistic culture in Alexandria that attracted many of the greatest mathematical scholars of antiquity, including Euclid, Apollonius, Archimedes, Eratosthenes, Hero, Pappus, and Claudius Ptolemy.


return to main Project MATHEMATICS! screen

E-mail your questions to Tom Apostol