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Which is the formal definition of the function $Sin$, starting from axioms of real numbers ?

I never found it in any book.

Not its Taylor serie, that is based upon the intuitive definition of $Sin$; I want the formal definition of $Sin$ without confusing drawings of circles, angles and straight lines.

halfpog
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5 Answers5

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One formal definition is via complex analysis, as used, e.g., by metamath (df-sin): $$ \sin z=\frac{e^{iz}-e^{-iz}}{2i}. $$

This, in turn, rests on the definition of $\exp$ (df-ef): $$ \exp z=\sum_{k=0}^\infty\frac{z^k}{k!}. $$

For more details see the definition of sum (df-sum) which, in the case of infinite sums, is defined in terms of limits (df-clim).

Charles
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All the definitions go in circles, in that from one you can always get to the other. Some people who are very big into foundations (e.g. Bourbaki) have specific preferences for what should be the "formal" definition.

To avoid a power series definition, we can first define the complex exponential function $\mathrm{cexp}(x)$ as the unique $2\pi$-periodic positively-oriented continuous group homomorphism $(\mathbb{R},+) \rightarrow (U, \cdot)$, where $(U, \cdot)$ is the complex multiplicative circle group (where $\pi$ can be defined by an integral). Then $\sin(x)$ and $\cos(x)$ are the imaginary and real parts of that function.

Christopher A. Wong
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    "All definitions go in circles"?? Cut out that suggestive language, and give me a huge pile of symbols to chew on! (but really, +1) – pjs36 Jun 05 '15 at 00:07
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The $\sin$ function is usually defined using "confusing drawings of circles, angles and straight lines." It is usually from this that you derive its Taylor series, which is another possible candidate for the definition. I don't believe there are any other ways of defining it and I don't see any connection to the "axioms of real numbers."

James
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  • My question is a bit unclear, I know. What I want to say is avoid circular definition or reasoning. It seems that the radians is defined using $\pi$, $\pi$ is calculated using $sin$ and radians, and $sin$ is defined using radians. For Taylor serie I have to derive $sin$, and so I have to define it before. I want a definition of $sin$ without any geometric intuition. – halfpog Jun 04 '15 at 14:55
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    $\pi$ need not be, and rarely is, defined/calculated using radians. It's simply the ratio of any circle's circumference to its diameter, as it has been since antiquity. – pjs36 Jun 04 '15 at 16:24
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If you wish, you can define $\sin$ by solution of differential equation $$\ddot x + x = 0,\enspace x(0)=0,\dot x(0)=1,$$ and $\cos$ in the same manner.

Michael Galuza
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Maybe I found an answer in this web page:

http://math2.org/math/algebra/functions/trig/sin-cos-definition.htm

halfpog
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