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