Studying Mathematics Thread

[Fireball]

In other words, if you multiply two expressions of the form cos(A)+i*sin(A), the result can be obtained by *adding* the angles involved.

Now, what happens if you multiply the *same* expression over and over again? The angle adds up again and again, however many times you did the multiplication.

That is why

[cos(A)+i*sin(A)]^n = cos(n A)+i*sin(n A)

Each multiplication corresponds to an addition of the angles.

So I took out paper and pencil and worked it out along with your example. I can clearly see how cosine and sine sum/difference formulas fit into DeMoivre's theorem. Sometimes I need it spelled out, but once I started following along with you, it was clear where the steps would take me. I was still scratching my head over how the exponent plays into this, but then it hit me like a ton of bricks. I see the little cycle there of adding and multiplying.

In short, you've effectively explained it to me, so thank you.

Watch out, or he'll have you studying analytical algebraic topology of infinitely differentiable Riemannian Manifolds! Boyja Moi!

Finally got a chance to post that. Special thanks go to Tom Lehrer for making a song about it!