Complex Numbers: Trig Identities: 1. De Moivre's Theorem states that for whole number n, (\cos \theta + i \sin \theta)^n = \cos n \theta + i \sin n \theta. You are interested in the case n = 3. Expand the binomial and equate real and imaginary parts. share|cite|improve this answer. answered Mar 23 '13 at Complex Numbers in Trig Identities II vincenthuang October 12, 1 Introduction. Trigonometry is generally considered bashy, and perhaps that is in.

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Work on the problem of general polynomials ultimately led to the fundamental theorem complex numbers identities algebrawhich shows that with complex numbers, a solution exists to every polynomial equation of degree one or higher. Complex numbers thus form an algebraically closed fieldwhere any polynomial equation has a root.

Many mathematicians contributed to complex numbers identities full development of complex numbers. The main point being that if you know complex multiplication, you never have to remember what the grouping of factors and the signs are, something I used to have trouble remembering.

So not complex numbers identities a new formula either! Moving either the cosine or sine terms to the right-hand side gives the two immensely useful equations: This is a little bit of a refresher, you can multiply these two together to verify you get X squared minus Y squared.

In fact, let's just do that for fun.

These two middle terms complex numbers identities out and your left with X squared minus Y squared. What I want to tackle in this video is something that we didn't know how to factor before and that's the complex numbers identities of the squares.

So if we were to factor X, let me use a brighter color, if we want to try to factor X squared plus Y squared, before we knew about imaginary numbers, complex numbers, we didn't know how to factor this.

Fancy meeting you here! Can we do something with the sine double-angle formula too?

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Using andwe get: And it is now called Euler's Formula. Let's give it a try: