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Posted November 19th, 2007 by Anonymous
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I want to learn more about the function sin(x).

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Comments

About sin function

On November 19th, 2007 Structure says:

Here are some properties of real and complex function called sin.
First of all let begin with a definition.
For a beginner we can start can start as the fraction associated with a right triangle.
$ sinx=\frac{opposite side}{hypotenuse} $
From Thales theorem this value depends only of x angle.
Unfortunately, this elementary definition is not enough for mathematics so we have to extend to the real axis.Taking the unit circle, a point of it (x,y) depends of a variable t and x=cost; y=sint, where t is the length of the arc sector.So we have $ \cos(t+2\pi)=\cos t \: ;\sin(t+2\pi)=\sin t $
$ \cos^2t+\sin^2t=1 $
$ \sin (x+y)=\sin x\cos y+\sin y\cos x $
$ \sin (-x)=-\sin x $
$ \sin x=a $ has a countable set of solutions if $ |a|\le 1 $
$ x_k=k\pi +(-1)^k \arcsin a $
$ (\cos x+i\sin x)^n=\cos nx+i\sin nx $ Moivre formula;
Now for analyst

$$\sin x = \sum _{n=0}^{+\infty}(-1)^n \frac{x^{2n+1}}{(2n+1)!}$$
$$\cos x=\sum_{n=0}^{+\infty}(-1)^n\frac{x^{2n}}{(2n)!}$$

Another things respecting to

On December 4th, 2007 Krizalid says:

Another things respecting to integrals, are

$$\int_0^\infty {\frac{{\sin x}}<br /> {x}\,dx} = \int_0^\infty {\frac{{\sin ^2 x}}<br /> {{x^2 }}\,dx} = \frac{\pi }<br /> {2}.$$

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