Solving a particular trigonometric equation I am wondering if it is possible to solve the equation
\begin{equation}
\sin(x) = 0.4.
\end{equation}
If it is possible to solve this, how does one do so?

Yes you can, if you plot the function $\sin x$ then you can see that what you want are the values of $x$ where $\sin x$ is 0.4. This is shown by the red line in the plot below.

First thing to note is that because $\sin$ is a periodic function, there is more than one solution, as witnessed by the fact that the red line hits the black line in more than one place. You usually choose the range that you want $x$ to be before you start, in this case, for $\sin$, between $-90$ and $90$ degrees is usual.
Second thing to note is that if you'd chosen a value for $\sin(x)$ outside the range $-1$ to $1$ then there's no way the red line could hit the black sin wave, and so there would be no solution.
Also, technically, to solve an equation like this we apply the inverse $\sin$ function, $\sin^{-1}$ (a.k.a. arcsin), to each side of the equation. Applying an inverse function to itself cancels out, so we have
$$\sin(x) = 0.4$$
$$\sin^{-1}(\sin x) = \sin^{-1}(0.4)$$
$$ x = \sin^{-1}(0.4)$$
you can look up the result on a calculator. (Apart from a few special values like 0 degrees, 30 degrees, 45 degrees, 60 degrees, 90 degrees and so on, the values of $\sin^-1$ aren't easily known and so the calculator has to follow a numerical algorithm to get you the answer.)
The answer is 23.6 degrees, to 1 decimal place, as shown by the green line.
