Studying Mathematics Thread

[polymath257]

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.

I'm really glad I could help!

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But wait... seriously... why do you just get to put the exponent in front of the cos and isin? How does that work? See, I once had this professor who insisted on proving to us why a formula worked, and I'd sit there in class thinking, "I don't give a shit! This is boring and confusing. Just give me the formula and I'll plug in my little values and get an A in your class. Cause that's what I do!"

But now I'm in this place where I'm seeing things and I can't just take DeMovire's word for it. I'll grant that I'm more inclined to take Euler's word for it, but I'd still like to know why this formula works.

Have you checked this link?

https://proofwiki.org/wiki/De_Moivre%27s_Formula

If that doesn't help, what do you mean by "exponent in front of the cos and isin"? Because, from what I see, only r has a variable exponent.

The biggest problem here is that the identification

exp( i*A)=cos(A)+i*sin(A)

is not at all easy to see. There are two main ways I know of: one uses the power series expansions of exp, sin, and cos. The other does it by solving a differential equation.

But DeMoivre's theorem is something that can be appreciated well before those topics are covered.