De Moivre's theorem
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Applications of De Moivre's Theorem to Trigonometric Identities
By expanding [Equation goes here - download the original to see it.] using the Binomial theorem (or Pascal's triangle) and equating with [Equation goes here - download the original to see it.] we can obtain further trigonometric identities. Recall that De Moivre's theorem is [Equation goes here - download the original to see it.] Since the real and imaginary parts of both sides of this equation are independent of each other, we can equate real and imaginary parts to obtain trigonometric identities. The whole process is best grasped through illustration [Equation goes here - download the original to see it.] Pascal's triangle up to n = 5 gives [Diagram goes here - download the original to see it.] Hence [Equation goes here - download the original to see it.] Since i2 = -1 we have [Equation goes here - download the original to see it.] On equating real and imaginary parts and using the identity [Equation goes here - download the original to see it.] [Equation goes here - download the original to see it.] Also [Equation goes here - download the original to see it.]
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Contents of
De Moivre's theorem
1 De Moivre's Theorem
2 Interpretation of De Moivre's Theorem and the n roots of unity
3 Applications of De Moivre's Theorem to Trigonometric Identities
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