De Moivres formula for multiple angles. Where does the sinus go?

[mariush]

Hi!

Yesterday i read and mostly understood that (cos([itex]\theta[/itex]) + isin([itex]\theta[/itex])[itex]^{n}[/itex] = (cos([itex]n\theta[/itex]) + isin([itex]n\theta[/itex])

 

[HallsofIvy]

Okay, so what is your question? What about the multiple angles that you mention in your title?

Note that the formula you give can be derived from the formula for multiplication of complex numbers in "polar form"- [itex]r_1(cos(\theta_1)+ i sin(\theta_1))r_2(cos(\theta_2)+ i sin(\theta_2))= (r_1r_2)(cos(\theta_1+ \theta_2)+ i sin(\theta_1+ \theta_2))[/itex].

Also, we can use Euler's formula, [itex]e^{ix}= cos(x)+ i sin(x)[/itex] to write [itex](cos(\theta)+ i sin(\theta))^n= cos(n\theta)+ i sin(n\theta)[/itex] in the simpler form [itex](e^{i\theta})^n= e^{i n\theta}[/itex].

 

[mariush]

Sorry, I'm not best friends with the post publisher today.

My question was about the multiple angles: I understand that
[itex](cos x +i sinx)^{n} = cos (nx) + i*sin(nx)[/itex]. Then i read that [itex]cos (nx) = cos^{n}(x) - (nC2)cos^{n-2}x*sin^{2}(x)- ...-(-1)^{k/2}(nCk)cos^{n-k}(k)*sin^{k}(x) [/itex]

Now, that looks a lot like what I would expect to get from [itex]cos (nx) + i*sin(nx) = (cos x +i sinx)^{n},[/itex] but i cannot figure out where the sin(nx) goes.

My first thought is that [itex]cos (nx) = (cos x +i sinx)^{n} - sin(nx)[/itex]

 

[mariush]

Okay, so what is your question? What about the multiple angles that you mention in your title?

Note that the formula you give can be derived from the formula for multiplication of complex numbers in "polar form"- [itex]r_1(cos(\theta_1)+ i sin(\theta_1))r_2(cos(\theta_2)+ i sin(\theta_2))= (r_1r_2)(cos(\theta_1+ \theta_2)+ i sin(\theta_1+ \theta_2))[/itex].

Also, we can use Euler's formula, [itex]e^{ix}= cos(x)+ i sin(x)[/itex] to write [itex](cos(\theta)+ i sin(\theta))^n= cos(n\theta)+ i sin(n\theta)[/itex] in the simpler form [itex](e^{i\theta})^n= e^{i n\theta}[/itex].

Hi, and thanks for such a quick response! I'm sorry about the first post. Seems like I hit the submit rather than the preview button, a bit prematurely..

 

[CellCoree]

).

De Moivre's formula for multiple angles is a powerful mathematical tool that allows us to find the value of a complex number raised to a multiple angle. This formula states that when a complex number, represented by cos(\theta) + isin(\theta), is raised to the nth power, the resulting complex number is cos(n\theta) + isin(n\theta).

But where does the sinus go in this formula?

The sinus, or the sine function, is represented by the imaginary part of the complex number (isin(\theta)). When raised to the nth power, it remains in the imaginary part of the resulting complex number (isin(n\theta)). This is because the sine function is periodic and has a period of 2\pi, meaning that when we raise it to a multiple angle, it will still have the same value in the imaginary part.

Overall, De Moivre's formula is a useful tool for simplifying complex number calculations and understanding the behavior of the sine function when raised to multiple angles.