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In physics, particularly in waves, we make use of the fact that for small angles (less than $\pi/12$-ish), the sine function value of an angle is pretty close to the value of the angle itself (in radians of course). Can anyone give a mathematical explanation for why this is?

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If you draw a triangle, and limit the angle $\theta$ to zero, i.e., make it smaller and smaller, you see that the opposite side is very small compared to the hypotenuse. Therefore, $\sin \theta\to0$ as $\theta\to 0$. –  Sanath Feb 9 at 16:26
    
You may want to see this question on the corresponding question for cosine: math.stackexchange.com/questions/113416/… –  Batman Feb 9 at 16:27

4 Answers 4

up vote 3 down vote accepted

Hint

Any mathematical function can be, at least locally, approximated by so called Taylor or Mc Laurin expansions.

To make it as simple as possible, the tangent to a curve is, at the point where it is defined, a local approximation of the curve.

So, write the equation of the tangent to the curve $y=sin(x)$ at $x=0$ and you will obtain, for the tangent line, $y = 0 + (x-0) = x$. So, close to $x=0$, $sin(x)$ is close to $x$.

Uisng the same approach, you could show by yourself that, close to $x=0$, $e^x$ is close to $1+x$, that $log(1+x)$ is close to $x$ and so on. For sure, the approximations can be made better and better at the price of more terms.

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Alternative solution, if you do not want to deal with series expansion, you could calculate

$$\lim_{x\to0} \frac{\sin x}{x} = 1\quad\text{and/or}\quad \lim_{x\to0}\frac{x}{\sin x}=1$$

Thus $\sin x \sim x$ for $x$ close to $0$.

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Hint

Consider the Taylor expansion of $\sin{x}$ about $x = 0$.

Cheers!

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Of course! But now I wonder, is there a way to angle this useful behavior without such series? –  Just_a_fool Feb 9 at 16:18
    
I was about to show you that the tangent curve of $\sin{x}$ about $x = 0$ is precisely $y(x) = x$, so $\sin{x} \sim x$ in the neighbourhood of $x \to 0$, but... @Claude Leibovici was quicker than me, haha! –  Dmoreno Feb 9 at 16:28

Indeed, there a way to angle this useful behavior without such series. In analysis involving real numbers $\sin x$ will always be appreciably different from $x$. However, in a framework involving an infinitesimal-enriched continuum such as the hyperreals the richer syntax allows one to formulate ideas such as the replaceability of $\sin x$ by $x$ for "small" $x$, where "small" is interpreted as infinitesimal. For example, the problem of small oscillations of the pendulum where one wishes to argue that the solution is basically given by harmonic motion, admits a rigorous formalisation in this setting. In particular, one can give a clean proof of the fact that the period is independent of the amplitude without needing to let anything tend to anything else.

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