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I am after a reference (book, papers etc) about the practical applications of trigonometric functions raised to higher powers.

An example is one that I have been using in my own studies:

$\cos^4 \theta$

I am interested in further examples of applications of these types of functions, particularly if they are trigonometric identities.

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1 Answer 1

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1) Two applications I can think of relate to Wallis integral : Considering the sequence $(I_{n})_{n \geq 0}$ defined by

$$ I_{n} = \int_{0}^{\frac{\pi}{2}} \sin^{n}(x) \: dx = \int_{0}^{\frac{\pi}{2}} \cos^{n}(x) \: dx $$

one can prove the Stirling formula (see http://en.wikipedia.org/wiki/Wallis%27_integrals) :

$$ n! \sim \left( \frac{n}{e} \right)^{n} \sqrt{2n\pi} $$

or the Wallis product (see http://en.wikipedia.org/wiki/Wallis_product#cite_note-2) :

$$ \prod_{n=1}^{+\infty} \frac{2n}{2n-1} \frac{2n}{2n+1} = \frac{\pi}{2} $$

-- Edit : another application of Wallis integral :

Let $R >0$ and $n \in \mathbb{N}^{\ast}$. Let $B = \lbrace x=(x_{1},\ldots,x_{n}) \in \mathbb{R}^{n}, \, \Vert x \Vert_{2} \leq R \rbrace$ be the unit ball of $\mathbb{R}^{n}$ and let $V_{n}$ be its volume. Using Wallis integral, one can prove that :

$$ V_{2n}=\frac{\pi^{n}}{n!} R^{2n} $$

and

$$ V_{2n+1} = 2^{2n+1} \frac{n!}{(2n+1)!} \pi^{n} R^{2n+1} $$

and

$$ \lim \limits_{n \rightarrow +\infty} V_{n} = 0 $$

2) Using Euler identities ($\cos(x)=\frac{e^{ix}+e^{-ix}}{2}$ and $\sin(x) = \frac{e^{ix}-e^{-ix}}{2i}$) and the binomial theorem (see http://en.wikipedia.org/wiki/Binomial_theorem), one can prove the following identities :

$$ \cos^{2n}(x) = \frac{1}{4^{n}} \left( \begin{pmatrix} 2n \\ n \end{pmatrix} + 2 \sum_{k=0}^{n-1} \begin{pmatrix} 2n \\ k \end{pmatrix} \cos \left( 2(n-k)x \right) \right) $$

and

$$ \cos^{2n+1}(x) = \frac{1}{4^{n}} \sum_{k=0}^{n} \begin{pmatrix} 2n+1 \\ n-k \end{pmatrix} \cos((2k+1)x) $$

(and there are similar identities for $\sin^{2n}(x)$ and $\sin^{2n+1}(x)$. These identities allow you to integrate powers of $\cos$ and $\sin$.

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+1, very nice answer - do you have any non-Wikipedia links? –  user83622 Aug 13 '13 at 19:35
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Have a look at this : - math.wisc.edu/~angenent/276/wallis.pdf - math.mcgill.ca/drury/notes255.pdf (p.125) - math.uwaterloo.ca/~krdavids/RAA/RAAsupp.pdf (p.36) If you read French : - gilles.costantini.pagesperso-orange.fr/agreg_fichiers/… - maths-france.fr/Capes/Capes_2009_M1_Enonce.pdf Unfortunately, I have no reference for the $\cos^{2n}(x)$ and $\cos^{2n+1}(x)$ identities. + I added another application of Wallis integral to my post. –  jibounet Aug 14 '13 at 8:47
    
Awesome! This is the information I am after. Thank you for this. –  user83622 Aug 14 '13 at 9:26
    
Thanks to you ! :-) –  jibounet Aug 14 '13 at 9:27

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