$$x\sin(x)=3$$
It seems so simple, but I'm not quite sure how to solve it besides plugging it into a graphing calculator. This was a question on a test, where the solution was simply to use the calculator.
Can it even be solved without a calculator?
What I can think of off the top of my head as a solution is to use a Taylor series to approximate $x\sin(x)$, which would turn it into a polynomial that could be solved algebraically. Is this, then, the only solution?
This is not quite the same as the old (closed) problem. For each $a$, the equation $x\sin(x)=a$ has solutions near $n\pi$ for every sufficiently large integer $n$. For example when $a=3$ as in this thread, we need only exclude $n=0,\pm 1$; for every other $n$, Newton's Method will converge quickly to a numerical solution of the OP's problem.
But this value $a=3$ is big enough to exclude another sort of solution that could have been offered in the old thread. Because as long as $0<a< 1.8197$ or so (the height of the first local max) it also makes sense to ask whether we can solve $x\sin(x)=a$ as a smooth function $x=f(a)$. The answer is yes, and although there's no handy "symbolic" expression for $f$, there is a power-series solution: $$f(a) = b + (1/12) b^3 + (29/1440) b^5 + (263/40320) b^7 + \dots$$ where $b$ is either of the square roots of $a$. (The coefficients are obtained by inverting the power series of $x\sin(x)$ and then extracting a square root. I don't see any particular pattern to them; they're rational and once multiplied by an appropriate factorial they become nearly integral; they all seem to be positive and decreasing at a rate that suggests the appropriate radius of convergence.)
Hint
Consider the function $$f(x)=x \sin(x)-3$$
If you use inspection for $x=k\frac \pi 4$, you find quickly that the first positive solution is between $k=8$ and $k=9$.
Just compute the equation of the line joining these two points and compute the $x$ intercept to obtain $$x\sim\frac{\sqrt{2}}{3} \left(1+3 \sqrt{2} \pi \right) = 6.75459$$ while the solution is $6.74417$.
