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I am trying to work out how to solve the following functions graphically.

On the same axes, draw the graph of $y = \sin x$ ($x$ in radians) and $y = \dfrac{1}{x}$ for values of $x$ between $0.5$ and $1.5$. Use graphs to estimate a value of $x$ such that $x\sin x = 1$.

Explain why when $x$ is large, solutions of the equation $x\sin x = 1$ are given approximately by $x = n\pi$, where $n$ is an integer.

How do I go about graphing different units on the same axes. I thought of using equivalent degrees but that makes the sin graph too large for values $0.5$ and $1.5$. Is this type of problem only for solving with calculators?

Thanks again for all your help.

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The problem specifically states that $x$ is in radians, so why would you switch to degrees? I think that whoever gave you this problem wants you to plot the two functions (BTW, fix the latter to read $y=1/x$). Very carefully in the prescribed range $0.5\le x\le 1.5$. Find an approximate point of intersection that should give one solution of the equation. And then to answer the final question you are to make a coarser plot of the functions for large values of $x$, and make some observations about the $x$-coordinates of the points of intersection. – Jyrki Lahtonen Jul 1 '11 at 6:18
Your previous question was also tagged under [elementary-set-theory]. Please explain, why do you think this has anything to do with elementary set theory? – Asaf Karagila Jul 1 '11 at 6:33
I assumed it applied to graphing of simple trigonometric functions. Please feel free to correct this if it's wrong. – mathguy80 Jul 1 '11 at 9:15
Thanks @Jyrki, it turned out to be simple after I plotted the points. – mathguy80 Jul 1 '11 at 9:16

I think you're confused about units. When $x=\pi$, say, the first function is $\sin\pi=0$, and the second function (assuming you really meant $y=1/x$) is $1/\pi$, and there's no problem plotting these on the same axes.

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Yes it's $y = \dfrac{1}{x}$. I have fixed the typo. I was confused about how to get the graph to be accurate for both the functions at the same time. As @Jyrki pointed out I have to plot points between $(0.5, 1.5)$. – mathguy80 Jul 1 '11 at 9:02
up vote 1 down vote accepted

I figured out this problem.

As x values become very large the graph of $y = \dfrac{1}{x}$ becomes just the 2 asymptotes at $y = 0$ and $x = 0$.

The graph of $y = \sin(x)$ intersects the horizontal asymptote at points $0, \pm\pi, \pm3\pi, \pm4\pi$, ... Thus for large values of x, the solutions of the equation $x\sin x = 1$ are at x = $0, \pm\pi, \pm3\pi, \pm4\pi$, ...

In other words, $x = n\pi$, where $n \in \mathbb{Z}$.

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the solutions are near those $x$-values, but not at them. $3\pi\sin3\pi\ne1$. – Gerry Myerson Jul 1 '11 at 13:08

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