Multiple Angle Formulas for Sine and Cosine

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Mathematica can directly produce formulas for the sines and cosines of multiple angles.


Here is a derivation of these formulas. Euler's formula is . Then, , but also, . The relation is known as de Moivre's theorem.

For the case , the right-hand side can be expanded to give


using . Equating the imaginary and real parts leads to the double angle formulas



The cases are similar, using a binomial expansion.


Contributed by: S. M. Blinder (July 2018)
Open content licensed under CC BY-NC-SA



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