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Explain the process of using de Moivre's Theorem to find a trigonometric identity. For example, express tan(3x) in terms of sin(x) and cos(x).

  1. Identify de Moivre's Theorem: (cos(x) + isin(x))n = cos(nx) + isin(nx) 2) Deduce the correct value of n for the given problem. In this example we set n=3 3) Expand the LHS (usually by a binomial expansion). In this example we have (cos(x) + isin(x))3 = cos3(x) + 3icos2(x)sin(x) - 3cos(x)sin2(x) - isin3(x) = cos(3x) + isin(3x) 4) Equate the real parts. Here we have cos(3x) = cos3(x) - 3cos(x)sin2(x) 5) Equate the imaginary parts. Here we have sin(3x) = 3cos2(x)sin(x) - sin3(x) 6) Use these results to derive identity. In this case we divide sin(3x) by cos(3x).
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