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I'm doing some exam review and I thought I might need to use something like this. Is it true? If not, is there a similar statement, and if so, how can we prove it? $$\sin\left(\frac1{n^k}\right) < \frac1{n^k} \forall n,k\in \mathbb{N}$$

I can't find it on the site or in my textbook, but it seems to be an assumption used quite a bit (or some variation of it). I'm assuming if it's true it can be improved to say for all real numbers greater than 1? Or possibly even better?

I tried checking on wolfram, but I'm not sure how to ask a question and make it accept conditions : http://www.wolframalpha.com/input/?i=is+sin%281%2Fn%29+%3C+1%2Fn

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4 Answers 4

up vote 7 down vote accepted

Consider $f(x)=\sin(x)-x$. Note that $f(0)=0$, and that $f'(x)=\cos(x)-1\le 0$, so $f$ is decreasing, i.e., $f(x)\le0$ for all $x\ge0$, or $\sin(x)\le x$. In particular, $$ \sin\left(\frac1{n^k}\right)\le\frac1{n^k}.$$

In fact, the inequality is strict, since otherwise $t=1/n^k$ satisfies $f(t)=0$. But then $f(0)=f(t)$ and, by the mean value theorem, there is an $s$ between $0$ and $t$ with $f'(s)=0$. But $\cos(x)<1$ for all $x\in(0,2\pi)$, and $1/n^k<2\pi$.

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+1, a very simple explanation, thanks. –  fdart17 Apr 17 '11 at 1:30
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It is true. First notice that if $0 < x < \pi/2$, we have $0 < \sin(x) < x$. This is true by the following argument. The point $(\cos(x), \sin(x))$ is the point you arrive at by traveling distance $x$ along the unit circle counterclockwise from $(1,0)$. Hence $\sin(x)$ is the vertical distance from $(\cos(x), \sin(x))$ to the $x$--axis. Since this vertical distance is the shortest path from $(\cos(x), \sin(x))$ to the $x$-axis, we have $0 < \sin(x) < x$ for $0 < x < \pi/2$.

Now take $x = 1/n^k$, and you are done.

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+1, a nice intuitive explanation. –  fdart17 Apr 17 '11 at 1:30
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$\sin(x) < x$ for all $x > 0$, which includes what you just said. The proof is using the derivative:

Let $f(x) = x - \sin x$. Then $f'(x) = 1 - \cos x \geq 0$ for all $x$. Since it's only equal to 0 at one point here and there, $2n\pi$ for $n \in \mathbb{Z}$, we know $f$ is increasing for all $x$. Since $f(0) = 0$, we see that $f(x) > 0$ for all $x > 0$, which is the same as $x > \sin x$.

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Numth, $\cos(\pi/2)=0$, not $1$. I think you mean $2\pi$, $4\pi$, etc. –  Andres Caicedo Apr 16 '11 at 23:12
    
@Andres Yes, thanks. I was thinking of where cos x was 0, not where 1 - cos x is 0. I will update it. –  Graphth Apr 17 '11 at 1:03
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Look at the function $\frac{\sin(x)}{x} $ on the interval $(0,1)$ and optimize by taking derivatives and such. You will find that $\frac{\sin(x)}{x}<1$ with the ''maximum'' attained when $x\to 0.$ Now with a little algebra you should get $\sin(x)<x.$

Now observe that $0<\frac{1}{n^k}<1$ for every $n,k\in \mathbb{N}.$ Thus, we can choose $x=\frac{1}{n^k}$ and you will get your desired inequality.

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