finding inverse of sin with a variable in the argument $$h(t) = 18\sin\left(\pi\frac{t}{7}\right) + 20$$
Where $h(t)$ is height in meters and $t$ is the elapsed time in seconds.
If $h$ is restricted to the domain $[3.5,10.5]$ find and interpret the meaning of $h^{-1}(20)$.
In the Facit the answer is $7$. This means that a height of $20$ meters is reached in $7$ seconds.

Let $h(t) = h$ and $t = h^{-1}$. In this case:
$h = 20 + 18sin(\frac{\pi h^{-1}}{7})$
$h - 20 = 18sin(\frac{\pi h^{-1}}{7})$
$\frac{h - 20}{18} = sin(\frac{\pi h^{-1}}{7})$
$arcsin(\frac{h - 20}{18}) = \frac{\pi h^{-1}}{7}$
$\frac{7 arcsin(\frac{h - 20}{18})}{\pi} = h^{-1}$
Now substitute 20 to $h$:
$h^{-1}(20) = \frac{7 arcsin(\frac{20 - 20}{18})}{\pi} = \frac{7 arcsin(0)}{\pi}$.
Given that $arcsin(0) = 0 + k\pi$, $\frac{arcsin(0)}{\pi}$ equals any natural number. Take all the natural numbers between 4 and 10, multiply each by 7. The least of them is 28, which leaves me with the thought that I've messed it up somewhere.
