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 Post subject: De Moivre question (complex numbers)
PostPosted: Sat, 23 Oct 2010 15:05:28 UTC 
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De Moivre's Theorem says that if \theta is real, then (cos\theta + isin\theta)^n = cos(n\theta) + isin(n\theta)for all n E Z.

By considering real parts, find an expression for cos(5\theta)in terms of powers of cos\theta and sin\theta.

Attempt:

cos(5\theta) = cos^2(2.5\theta) - sin^2(2.5\theta)

but this wasnt the correct answer.


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 Post subject: Re: De Moivre question (complex numbers)
PostPosted: Sat, 23 Oct 2010 15:17:13 UTC 
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TsAmE wrote:
De Moivre's Theorem says that if \theta is real, then (cos\theta + isin\theta)^n = cos(n\theta) + isin(n\theta)for all n E Z.

By considering real parts, find an expression for cos(5\theta)in terms of powers of cos\theta and sin\theta.

Attempt:

cos(5\theta) = cos^2(2.5\theta) - sin^2(2.5\theta)

but this wasnt the correct answer.


Read the question again. What can you tell me about (\cos\theta+i\sin\theta)^5?

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\begin{aligned}
Spin(1)&=O(1)=\mathbb{Z}/2&\quad&\text{and}\\
Spin(2)&=U(1)=SO(2)&&\text{are obvious}\\
Spin(3)&=Sp(1)=SU(2)&&\text{by }q\mapsto(\mathop{\mathrm{Im}}\mathbb{H}\ni p\mapsto qp\bar{q})\\
Spin(4)&=Sp(1)\times Sp(1)&&\text{by }(q_1,q_2)\mapsto(\mathbb{H}\ni p\mapsto q_1p\bar{q_2})\\
Spin(5)&=Sp(2)&&\text{by }\mathbb{HP}^1\cong S^4_{round}\hookrightarrow\mathbb{R}^5\\
Spin(6)&=SU(4)&&\text{by the irrep }\Lambda_+\mathbb{C}^4
\end{aligned}


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 Post subject:
PostPosted: Sun, 24 Oct 2010 01:29:44 UTC 
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Okay, I want to have a stab at this one:

\\
\cos (5 \theta) + i \sin(5 \theta) = (\cos(\theta) + i \sin(\theta))^{5} \\

= \cos ^{5} (\theta) + 5 i \cos ^{4} (\theta) \sin (\theta) + 10 i^{2} \cos ^{3} (\theta) \sin ^{2} (\theta) + 10 i^{3} \cos^{2} (\theta) \sin^{3} (\theta) + 5 i^{4} \cos(\theta) \sin^{4} (\theta) + i^{5} \sin ^{5} (\theta) \\

= \cos^{5}(\theta) + 5i \cos^{4}(\theta) \sin(\theta) - 10 \cos^{3}(\theta) \sin^{2}(\theta) - 10i \cos^{2}(\theta) \sin^{3}(\theta) + 5 \cos(\theta) \sin^{4}(\theta) + i \sin^{5}(\theta)

Now we want the real parts, which is everything without an i in it. So
\cos(5 \theta) = \cos (\theta) ( \cos ^{4} (\theta) - 10 \cos^{2}(\theta) \sin^{2}(\theta) + 5 \sin^{4}(\theta) )

Then, ofc, as \sin^{2}(\theta) = 1 - \cos^{2}(\theta) you could write it as a sum of cosines, but if your answer is in sines and cosines then the above should be fine :D

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 Post subject:
PostPosted: Sun, 24 Oct 2010 13:47:50 UTC 
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Thanks a lot. For the last question it asked writesin(5\theta) in terms of powers of sin\theta.

In this case how would this be possible since sin(5\theta) is imaginary?


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 Post subject:
PostPosted: Sun, 24 Oct 2010 14:24:55 UTC 
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For \sin(5 \theta) take all of the imaginary terms in the expansion of (\cos(\theta) + i \sin(\theta))^{5}. Then use \cos^{2}(\theta) = 1 - \sin^{2}(\theta) to rewrite the terms with cosines in them

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 Post subject:
PostPosted: Sun, 24 Oct 2010 15:55:38 UTC 
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TsAmE wrote:
Thanks a lot. For the last question it asked writesin(5\theta) in terms of powers of sin\theta.

In this case how would this be possible since sin(5\theta) is imaginary?


It's not imaginary, it's the imaginary part of the complex number. If you don't get it you should remember that x+iy is complex, but the point is that y itself is real.

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