The link between complex numbers and trigonometry

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Since a complex number is simply an alternative representation of a point in the Cartesian plane, which in turn defines an angle and a length, it is not surprising that complex numbers can be used to derive trigonometric identities.

We can derive an expression for example for cos(5x) by using the composite angle identity for cos(A+B) repeatedly, but it is far easier to use the expression z = cosx + isinx and de Moivre’s theorem which states that the nth power of z is cos(nx) + isin(nx).

Hence cos(5x) is the real part of the expansion of (cos(x) + isin(x)) to the nth power. Use the binomial expansion to help you evaluate the coefficients and ignore odd powers of i since these are complex.

What if you want to derive an expression for an nth power of cos(x)? We can do this too. Using de Moivre’s theorem again, we can show that if z = cos(x) + isin(x) then z + 1/z = 2cos(x) so for example the fifth power of cos(x) is identical to (z + 1/z)^n = 32cos^5(x). The left hand side expands as z^5 + 5z^3 + 10z + 10z^-1 + 5z^-3 +z^-5. Group these terms together to make 2cos(5x) + 10cos(3x) + 20cos(x).

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