Difficult equation question

2014-05-21

An answer to this question on the Mathematics Stack Exchange.

Question

I have the equation $ t\sin (t^2) = 0.22984$. I solved this with a graphing calculator, but is there any way to solve for $ t$ without graphing?

Thanks!

Answer

Using a Taylor series, $\sin(x)$ can be written as

$\sin(x)\approx x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \ldots$

Replacing $x$ for $t^2$ gives:

$\sin(t^2)\approx t^2 - \frac{t^6}{3!} + \frac{t^{10}}{5!} - \frac{t^{14}}{7!} + \ldots$

Plugging this into your original equation gives:

$t\sin(t^2)=0.22984$

$=t^3 - \frac{t^7}{3!} + \frac{t^{11}}{5!} - \frac{t^{15}}{7!} + \ldots=0.22984$

So you can see why solving this in a closed-form sense might be difficult.

That said, it's reasonable to think that there might be a value of $t$ less than one, in which case you can try neglecting the higher level terms (this is the small angle approximation).

This gives

$t^3=0.22984$

$t=0.61255046092664577035$

Graphically, you find a root at $\sim0.617544$. The difference is 0.8%.