1.

Sines and chord lengths  
2.

Addition formula for sines  
3.

Ptolemy's theorem on quadrilaterals  
4.

Applications of the addition formulas  
5.

Sines and cosines of special angles  
6.

Application to simple harmonic motion  
7.

Preview of part IV  
8.

AlexandriaCenter 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.