I have this question from my textbook, however I keep getting "unidentified", which is not the answer at the back. I was wondering what I'm doing wrong.
The question is: Given that in a water wheel the height (in meters) of a nail above the surface of the water is, as a function of time (in seconds), $h(t) = -4\sin(\frac{\pi}{4}(t-1))+2.5$, during what periods of time is the nail below the water in the first 24s of the wheel rotating?
What I tried was:
first i got the period $\begin{align*} \text{period} &= 2\pi \times 4/\pi = 8\\ &-4\sin\left(\frac{\pi}{4}\right)(t-1) + 2.5\\ -2.5/-4 &= \sin(\pi/4)t - \sin(\pi)\\ (\pi/4)t &= \sin^{-1}(-2.5/4 + \sin(\pi/4))\\ &= \text{unidentified..} \end{align*}$
normally after that I usually get a answer find the quadrant its in find the actuate angle then two possible angles(then more from adding the period), however i got unidentified
the answer in the textbook is $1.86s \lt t \lt 4.14s$ , $9.86s \lt t \lt12.14s$, $17.86 s\lt t\lt 20.14s$