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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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    Thanks @Jyrki, it turned out to be simple after I plotted the points.2011-07-01

2 Answers 2

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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. $3\pi\sin3\pi\ne1$.2011-07-01
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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 $(0.5, 1.5)$.2011-07-01