Difficult equation question

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 1

The best you can hope for in this situation is to be able to calculate the solution out to as many digits as you are asked for. There are numerical methods to do just that, for instance Newton's Method.

Answer 2

As is said, there is no closed form solution to this equation. No formula if you prefer.

In such cases, numerical methods are used, which means that different values for $t$ are tried, using specific strategies to get closer and closer to the solution.

It is useful to carry out the study of the function to get a rough idea where the solutions can be.

In this case, if you rewrite as $sin(p)=\frac{0.22984}{\sqrt p}$, to make more familiar functions appear, you see that you intersect a sinusoid with a kind of hyperbola, having an horizontal asymptote.

The first root can also be estimated from $sin(p)\approx p$ (for small $p$), so that $t\sin(t^2)\approx t^3$, and $t\approx\sqrt[3]{0.22984}=0.61255$. (Check: $0.61255\sin(0.61255^2)=0.22448$).

For large $p$, the equation becomes $sin(p)\approx 0$, meaning that you will find infinitely many solutions close to $p=k\pi$, i.e. $t=\sqrt{k\pi}$.

A yet better approximation is obtained by setting $p=q+k\pi$ (for small $q$), so that $sin(q+k\pi)\approx \pm q\approx\pm\frac{0.22984}{\sqrt{k\pi}}$ ($+$ for even $k$, $-$ for odd), i.e. $p\approx(-1)^k\frac{0.22984}{\sqrt{k\pi}}+k\pi$, and $t\approx\sqrt{(-1)^k\frac{0.22984}{\sqrt{k\pi}}+k\pi}$.

There could be additional solutions in between.

Answer 3

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%.